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THREE DIMENSIONAL GEOMETRY

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					      CHAPTER

                10
                     THREE-DIMENSIONAL
                     GEOMETRY
                THEME: History



                     H    istory is a record of significant facts, events, and people. The study of
                          history includes architecture, politics, music, population, language,
                     religion, and mathematics. Specifically, the history of mathematics stretches
                     to every part of the world and spans over thousands of years.
                     Three-dimensional geometry has been studied since ancient times.
                     Consequently, people from every culture have required three-dimensional
                     geometry to design shelters, clothing, and storage containers.
                      • Urban planners (page 431) supervise the construction of buildings,
                        roads, bridges, and other structures within a city. Because it is important
                        to maintain historically significant
                        elements, while providing for the needs
                        of a city, history plays its part in a city’s
                        development.
                      • Exhibit designers (page 451) create
                        images and displays that show a product,
                        historical event, or structure that is
                        pleasing to one’s eye.




                     mathmatters2.com/chapter_theme
418
Mathematicians throughout History
Mathematician                   Life span               Accomplishment

Archimedes                    c.298 –212 B.C.           Greatest mathematician of ancient times.
                                                        Made contributions in geometry.
Easley, Annie                 1933–                     Developed computer code used to identify
                                                        energy conversion systems for NASA.
Euclid                        c.325 –265 B.C.           Wrote The Elements, 13 books covering
                                                        geometry and number theory.
Gauss, Carl Friedrich         1777–1855                 Published treatise on Number Theory.
                                                        Discovered the asteroid Ceres.
Hypatia                       c.370–415                 She is considered the first woman of
                                                        mathematicians.
Khayyam, Omar                 c.1048–1131               Persian mathematician, astronomer and
                                                        poet. Revised the Arabic calendar. Solved
                                                        cubic equations through geometry.
Lovelace, Ada Byron           1815–1852                 Suggested that Babbage's first "computer"
                                                        calculate and play music.
Pascal, Blaise                1623–1662                 Invented and sold the first adding machine
                                                        in 1645. Developed probability theory.
Pythagoras of Samos           c.560 –c.480 B.C.         His theorem showed the existence of
                                                        irrational numbers.

(c. stands for circa, meaning about or approximately)




Data Activity: Mathematicians throughout History
          Use the table for Questions 1–4.
          1. How long did Pythagoras live?
          2. Who is considered the greatest mathematician of ancient times?
          3. How many of these mathematicians did work with number
             theory? Who were they?
          4. List the names of mathematicians in the table with whom you
             are familiar. Discuss with a partner one or two other
             accomplishments credited to each person on your list.
                                                                                               Gauss

   CHAPTER INVESTIGATION
          Many people have contributed to the modern understanding of
          geometry. By studying the work of early mathematicians, you can
          develop a better understanding of mathematical concepts.

         Working Together
          Design a timeline of significant mathematicians who made
          contributions to geometry. Use the timeline to understand how
          mathematics influenced life throughout history. Use the Chapter
          Investigation icons to guide you to a complete timeline.                             Pascal



                                                                     Chapter 10 Three-Dimensional Geometry   419
          10                   Are You Ready?
CHAPTER




                               Refresh Your Math Skills for Chapter 10


           The skills on these two pages are ones you have already learned. Use the
           examples to refresh your memory and complete the exercises. For additional
           practice on these and more prerequisite skills, see pages 576–584.

PERIMETER FORMULAS
           In this chapter you will work with three-dimensional solids. Formulas that apply to
           two-dimensional shapes can be a foundation for understanding solid geometry.

             Perimeter of a square              Perimeter of a rectangle                                            Perimeter of any polygon
             P      4s, where s       side      P 2l         2w, where l                    length                  P sum of the lengths of
                                      length    and w        width                                                  the sides


      Examples
           Find the perimeter of each figure.
                                                                                    7 ft              8 ft
                                          P    2(8)     2(5)                                                            P     4      7   8       6    9
                                          P    16     10                     4 ft                                       P     34 ft
                                   5 cm

                                          P    26 cm                                                         6 ft
                    8 cm                                                                   9 ft
           Find the perimeter of each figure.
           1.                                         2.             5 in.                                              3.
                                                                                           8 in.
                 7 cm                                      7 in.                                                             12 m                15 m

                                                                                                   5 in.
                             12 cm
                                                                                                                                         18 m
                                                               12 in.
                                                                                           9 in.


           4.                                         5.     6.5 m              6.5 m                                   6.               18 yd

                 22 cm                                                                                                       11 yd                    9 yd
                                                           6.5 m                     6.5 m
                                                                                                                                         24 yd
                           22 cm                                        6.5 m


AREA FORMULAS
          Area of a rectangle or square               Area of a parallelogram                                       Area of a triangle
                                                                                                                             1
          A l w, where l             length           A b h, where b length                                         A          bh, where b           length
                                                                                                                             2
          and w width                                 of the base and h height
                                                                                                                    of the base and h                height

420       Chapter 10 Three-Dimensional Geometry
 Examples
     Find the area of each figure.
                                                                                                                 1
                                            A     15 7                                                    A        (20)(15)
              7 mm                                                                                               2
                                                                                 15 in.
                                                                                                                 1
                                            A     105 mm2                                                 A        (300)
                     15 mm                                                                                       2
                                                                                                          A      150 in.2
                                                                                   20 in.
     Find the area of each figure.
      7.                                                  8.                                               9.                23 cm

            4 in.                                              13 ft
                                                                                                                16 cm

                           5 in.
                                                                          13 ft


     10.                                                 11.                                              12.
                            12 in.
                                                                                                                        15 cm
                                                                       32 m
                    17 in.
                                                                        45 m                                               9 cm

VOLUME FORMULAS
      Volume of a rectangular prism                                                Volume of a triangular prism
      V B h, where B area of the                                                   V B h, where B area of the
      base and h height                                                            base and h height


 Examples
     Find the area of the base and substitute that measure to find the volume.
                                     B      3 4                                                      1
                                                                                                 B     (5)(8)
                                                                                                     2
                      8 cm           B      12                                                       1
                                                                                                 B     (40)
                                     V      12 8                                                     2
                                                                                          9 ft
                     3 cm
                                                                                                 B   20
                                                   3
                                     V      96 cm
           4 cm                                                                  5 ft            V   20 9        180 ft3
                                                                               8 ft
     Find the volume of each figure.
     13.                                                 14.                                              15.
                                     9 yd                                          10 m
                                                               8m                                             12 mm
                                                                          13 m
                                   3 yd                                                                                                mm
                    8 yd                                                                                                          22
                                                                                                                    9 mm

                                                                                                     Chapter 10 Are You Ready?              421
10-1                                 Visualize and
                                     Represent Solids
                                     Goals           I
                                                     I
                                                         Identify properties of three-dimensional figures.
                                                         Visualize three-dimensional geometric figures.
                                     Applications    Packaging, History, Sports, Machinery, Recreation


        Use a pencil, paper, tape and scissors.
        1. Cut a small rectangle, a small right triangle and a small semicircle out of
           the paper.
        2. Tape the rectangle near the top of the pencil. Spin the pencil on the tip to
           visualize what solid is created by spinning the rectangle. Describe the solid.
        3. Repeat Question 2 for the right triangle.
        4. Repeat Question 2 for the semicircle.



      BUILD UNDERSTANDING
        Some three-dimensional figures have both flat and curved                              Axis               Axis
        surfaces.

        A cylinder has a curved surface and two congruent circular                Right                  Oblique
        bases that lie in parallel planes. The axis is a segment that            cylinder                cylinder
        joins the centers of the bases. If the axis forms a right angle with
         the bases, it is a right cylinder. If not, it is an oblique cylinder.
                                                                                                      Axis              Axis
        A cone has a curved surface and one circular base. The axis is a
        segment that joins the vertex to the center of the base. If the axis                                 Oblique
                                                                                              Right
        forms a right angle with the base, it is a right cone. Otherwise, it is               cone            cone
        an oblique cone.
                                                                                       Center

        A sphere is the set of all points that are a given distance
        from a given point, called the center of the sphere.

        A polyhedron (plural: polyhedra) is a closed, three-
        dimensional figure made up of polygonal surfaces.                    Sphere
        The polygonal surfaces are faces. Two faces meet, or                                  Face               Base
        intersect, at an edge. A point at which three or more                          Edge
        edges intersect is a vertex.
                                                                                  Vertex
        A prism is a polyhedron with two identical parallel faces called
        bases. The other faces are parallelograms. A prism is named                   Base
                                                                                                             Pentagonal
                                                                                                               prism
        according to the shape of its base.
                                                                Triangular        Vertex
        A pyramid is a polyhedron with only one base.            pyramid
        The other faces are triangles that meet at a vertex.                           Face
        A pyramid is named by the shape of its base.

                                                                                        Base


422    Chapter 10 Three-Dimensional Geometry
     In prisms and pyramids, the faces that are not bases are lateral
     faces. The edges of these faces are lateral edges. Lateral faces can be            Check
     either parallel or intersecting. The lateral edges can be intersecting          Understanding
     or parallel. Skew edges are noncoplanar.
                                                                                   Why is a cube also a
                                                                                   rectangular prism?




       Right                        Oblique       Right              Oblique
    rectangular                   rectangular    square               square
       prism                         prism      pyramid              pyramid


Example 1
     PACKAGING Identify each feature of the shoe box.                                                    B    C

     a. the shape of the shoe box               b. a pair of bases
                                                                                     A
     c. a pair of parallel edges                d. a pair of intersecting edges                      D        H
                                                                                          Style A5       G
                                                                                           Brown
     e. a pair of skew edges                    f. a pair of intersecting faces            Size 9

                                                                                      E              F

 Solution
     a. The shoe box is a right rectangular prism.          b. Two bases are ADFE and BCHG.
     c. DC and FH are parallel edges.                       d. GH and FH are intersecting edges.
     e. AD and CH are skew edges.                           f. ABCD and CDFH are intersecting faces.


Example 2
     Draw a right triangular prism.

 Solution
     Step 1 Draw two congruent triangles.
     Step 2 Draw segments that connect the corresponding vertices
            of the triangles. Use dashed segments to show the edges
            that cannot be seen.


  TRY THESE EXERCISES
     Identify each figure and name its base(s).
     1.   A               B                                 2.           S


                              F
                      E
          D               C
                                                                 T             U

                  H           G


                                                                         V


       mathmatters2.com/extra_examples                    Lesson 10-1 Visualize and Represent Solids         423
        If possible, identify a pair of parallel faces, parallel edges,
        intersecting faces, and intersecting edges for each figure.
          3. the figure in Exercise 1                          4. the figure in Exercise 2

        Draw each figure.
          5. right circular cone                               6. oblique rectangular prism
          7. right hexagonal prism                             8. right square pyramid


      PRACTICE EXERCISES           • For Extra Practice, see page 616.
        Identify each figure and name its base(s).
          9.        E                       10.           A          B             11.               G       L

                                                                                                 H               K
                                                           D                                 A
                                                                      C                          F
                              G                                                                          I
                     F                            E                                                          J
                                                              F
                                                                                         B           E

               D         H                        H        G
                                                                                             C   D

        Identify a pair of parallel faces, parallel edges, intersecting faces, and
        intersecting edges for each figure.
        12. the figure in Exercise 9        13. the figure in Exercise 10          14. the figure in Exercise 11

        Draw each figure.
        15. right rectangular prism         16. oblique square pyramid             17. oblique cone
        18. right cylinder                  19. sphere                             20. right triangular prism

        21. WRITING MATH Describe ways in which prisms and pyramids are similar.
            Describe differences between prisms and pyramids.

        RECREATION If possible, identify the geometric figure represented and give the
        number of faces, vertices and edges.
        22. tennis ball can                 23. six-sided number cube              24. basketball
        25. ice cream cone                  26. rectangular gym                    27. one-person tent
        28. Which of the figures in Exercises 22–27 are polyhedra? Explain.

        MACHINERY A cutting drill uses different bits to cut designs into furniture.
        Describe the geometric figure cut when using each of the following bits.
        29.                                 30.                                    31.




        32. YOU MAKE THE CALL Niki says that a right square pyramid is a polyhedron
            but an oblique square pyramid is not a polyhedron. Daniella says both figures
            are polyhedra since their faces are polygonal. Who is correct and why?

424     Chapter 10 Three-Dimensional Geometry
  Decide whether each statement is true or false. Explain.
  33. A cube is a polyhedron.                            34. A polyhedron can have exactly three faces.
  35. A polyhedron can have a circular face.             36. A polyhedron can have exactly four faces.

  State whether each solid is a polyhedron. Explain.
  37.                             38.                              39.




  40. HISTORY The pyramids of Giza in Egypt are one of the seven wonders of the
      ancient world. Use the photo on page 433 to sketch the pyramid on paper.
      Then label the vertices.


EXTENDED PRACTICE EXERCISES
  SPREADSHEET Create a spreadsheet for the following figures. List figures in
  Column A, the number of faces (F ) in Column B, the number of vertices (V ) in
  Column C, and the number of edges (E ) in Column D. In Column E, write a
  formula to calculate the quantity F V E .
  41. rectangular prism                                  42. hexagonal prism
  43. pentagonal pyramid                                 44. square pyramid
  45. Compare your results for Exercises 41–44. Make a generalization about the
      relationship between the numbers of faces, edges and vertices in a
      polyhedron.

  46. CHAPTER INVESTIGATION Research ten mathematicians. Record the
      name, accomplishment, dates of birth and death, date of accomplishment
      and country of birth for each mathematician.


MIXED REVIEW EXERCISES
  Find the slope of the line containing the given points. Identify any vertical or
  horizontal lines. (Lesson 6-2)
  47. (4, 2), (6, 1)           48. (9, 3), ( 2, 4)       49. ( 1, 3), (2, 8)       50. (5,     4), (7, 7)
  51. ( 3,    5), ( 3,   2)    52. (4,   5), ( 2, 3)     53. ( 2, 6), (4, 6)       54. ( 1,      1), ( 7,       7)

  In the figure, RS MN and m 4           80°. Find each measure. (Lesson 5-3)
  55. m 1                         56. m 2
                                                                               R         7 8                S
  57. m 3                         58. m 5                                               5 6
  59. m 6                         60. m 7                                               3 4
                                                                               M       1 2              N




   mathmatters2.com/self_check_quiz                    Lesson 10-1 Visualize and Represent Solids               425
10-2                                 Nets and
                                     Surface Area
                                     Goals          I Draw nets for three-dimensional figures.
                                                    I Use nets to find the surface area of polyhedra.
                                     Applications   Retail, History, Machinery, Architecture


        Use grid paper and scissors.
        1. Copy the figure shown on grid paper, cut it out and fold along the
           dotted lines.
        2. What type of polyhedron is formed?
        3. Draw a different arrangement of the six squares so when you cut and
           fold the figure it makes the same solid as the one created in Question 2.
        4. Test the figure drawn in Question 3.
        5. Discuss with your classmates those figures that did create the same solid and
           those that did not.


      BUILD UNDERSTANDING
        A net is a two-dimensional pattern that can be folded to form a three-dimensional
        figure. A three-dimensional figure can have more than one net.


Example 1
        What three-dimensional figure is represented by each net?
        a.                                b.                                      c.




 Solution
        Visualize each net being folded to form a three-dimensional figure.
        a. This figure will have four rectangular sides and two parallel square bases. It is
           a rectangular prism.
        b. This figure will have three triangular sides and a triangular base. It is a
           triangular pyramid.
        c. This figure will have three rectangular sides and two parallel triangular bases.
           It is a triangular prism.


426    Chapter 10 Three-Dimensional Geometry
Example 2
                                                                                              Problem Solving
     Draw a net for each three-dimensional figure.                                                  Tip
     a. right square pyramid                                                                 If you are unable to
                                                                                             figure out the three-
     b. right cylinder                                                                       dimensional figure a net
                                                                                             represents, copy the net,
                                                                                             cut it out and fold it to
 Solution                                                                                    find the answer.

     a.                                   b.




     Nets can be used to calculate the surface area of certain
     three-dimensional figures. The surface area of a figure is the
     sum of the areas of all its bases and faces.



Example 3
     RETAIL Find the minimum amount of gift wrapping paper
     needed to cover a box with dimensions 1 ft by 2 ft by 3 in.

 Solution
                                                                                                                 3 in.
     Draw the net for the rectangular prism, and calculate the
     areas of the six sides. Use the formula A l w.
     The top and bottom of the box are rectangles with
     dimensions 1 ft by 2 ft.
                                                                                                                         2 ft
            l w     l w                            Find the combined surface area.
            2 1     2 1        4 ft2
     The left and right sides of the box are rectangles with
     dimensions 2 ft by 3 in.
                     1 ft
            3 in.              0.25 ft             Convert 3 in. to feet.
                                                                                                         1 ft
                    12 in.
            l w l w                                Find the combined surface area.
            2 0.25 2 0.25              1 ft2
                                                                                                  Check
     The front and back of the box are rectangles with dimensions 1 ft                         Understanding
     by 3 in. Use 0.25 ft for 3 in.
                                                                                             In Example 3, why are
            l w l w                                                                          you asked to find the
                                               2
            1 0.25 1 0.25              0.5 ft                                                minimum amount of
                                                                                             wrapping paper needed
     The total surface area is the sum of the areas of the sides.                            for the box?
            4 + 1 + 0.5      5.5 ft2
     So the box will require at least 5.5 ft2 of gift wrapping paper.

      mathmatters2.com/extra_examples                                       Lesson 10-2 Nets and Surface Area       427
      TRY THESE EXERCISES
        Identify the three-dimensional figure for each net.
        1.                    2.                              3.                  4.




        Draw a net for each three-dimensional figure.
          5. cube                          6. pentagonal prism                   7. rectangular prism

        Draw a net for each figure. Then find the surface area. Round each answer to
        the nearest tenth. Use 3.14 for π.
          8.                               9.                                   10.
                                                                       4.2 ft
                                                                                                  23.5 cm
               0.5 m               4m
                       2m
                                                                   10 ft
                                                3 ft                                    11.1 cm

                                                       3 ft




      PRACTICE EXERCISES
      PRACTICE EXERCISES           • For Extra Practice, see page 616.

        Identify the three-dimensional figure for each net.
        11.                                12.                                  13.




        Draw a net for each three-dimensional figure.
        14. octagonal prism                15. triangular prism                 16. right cone

        17. HISTORY Design and draw a Native American tepee. Then draw a net for the tepee.
        18. Draw a net of a cube that has a side length of 6 m. Then find the surface area of the cube.
        19. ARCHITECTURE What is the total surface area of the glass                                            5 ft
            (including the roof) used to build the greenhouse? Draw a net
                                                                                        3 ft
            of the walls and the roof to solve the problem.


                                                                                 6 ft
                                                                                                            10 ft
428     Chapter 10 Three-Dimensional Geometry                                                  8 ft
  Find the area of each net. Use 3.14 for π. Round to the nearest tenth if necessary.
  20.                                              21.         5 ft                                22. The height of each triangle
                                                                                                          5 3
                                                                                                       is     .
                                                                                                           3
                                           4 cm
                                                                                  8 ft
                                  4 cm                        15.7 ft                                                             3 1 ft
                                                                                                                                    3




  23. WRITING MATH How can nets help you find the surface area of three-
      dimensional figures? What kinds of mistakes do you make when calculating
      surface area?

EXTENDED PRACTICE EXERCISES
  Use nets to find the surface area of each prism. Use 3.14 for π.
  24.                         12 in.              25.                         7m                        26.
                                                                                                                 2.5 ft

                                         9 in.
                                                                                                                                 10 ft

        7 in.         6 in.     10 in.
                                                        3m
                    8 in.



  27. MACHINERY The wheel of a steamroller is a cylinder.
      If the wheel’s diameter is 5 ft and its width is 7.2 ft,
      approximately how much area is covered by one
      complete wheel revolution? Use a net to visualize
      and solve the problem. Round the answer to the
      nearest whole number.
  28. CRITICAL THINKING There are 11 different nets for
      a cube. Draw as many of them as you can.
  29. Refer to exercise 28. To construct 20 cubes out of
      construction paper, which of the 11 nets would
      you use? Explain.

MIXED REVIEW EXERCISES
  Simplify each numerical expression. (Lesson 2-2)
  30. 4         3     6 4                          31. 8 6          42        3                         32. 2       (5     2)2       4
  33. 15        3       (7      3) 2               34. 4      8(3        4)         2                   35. (6       1)2    15 3            4

  Evaluate each expression when c                       11. (Lesson 2-2)
  36. 18        c                      37. 3(c    8)                  38. (c             8)2   4                39. 9       c(3c           2)

  Simplify. (Lesson 9-2)
  40. (4r) (9p)                                    41. (6r 2) (4r 3g 2)                                 42. ( 3a 2) (5ab 2)2
  43. (5c 3d) (2cd 4)                              44. ( 4k 4f 2) ( 6kf 6)                              45. (3s 3t 2) ( 6st 3q)

   mathmatters2.com/self_check_quiz                                                      Lesson 10-2 Nets and Surface Area                      429
Review and Practice Your Skills
PRACTICE          LESSON 10-1
      Choose a word from the list to complete each statement.
               ?__
        1. A ___ has one square base and triangular faces.
                                                                               a. cone
               ?__
        2. A ___ has a curved surface and one circular base.
                                                                               b. cylinder
               ?__
        3. A ___ has two congruent circular bases that lie in parallel
                   planes between a curved surface.                            c. sphere

        4. A ___ is a figure consisting of the set of all points in space
               ?__                                                             d. square pyramid
                   that are a given distance from a given point.               e. triangular prism
               ?__
        5. A ___ has a triangular base and triangular faces.                   f. triangular pyramid
               ?__
        6. A ___ has two triangular bases parallel to each other with
                   the faces that are parallelograms.

      Identify each figure. State the number of faces, vertices and edges of each.
        7.                                 8.                                 9.




PRACTICE          LESSON 10-2
      Identify the three-dimensional figure for each net.
      10.                                 11.                                12.




      Draw a net for each three-dimensional figure.
      13.                                 14.                                15.
                                                4 cm        12 cm


                         6.5 cm                  3 cm                                                4 in.
                                                                      5 cm         6 in.
                4 cm                                                                         3 in.



      Find the surface area of each figure.
      16. the figure in Exercise 13
      17. the figure in Exercise 14
      18. the figure in Exercise 15

430   Chapter 10 Three-Dimensional Geometry
PRACTICE        LESSON 10-1–LESSON 10-2
     Draw a net for each three-dimensional figure. (Lesson 10-2)
     19.                                 20.                                 21.




     22. Draw a triangular pyramid.
     23. How many vertices, faces and edges does a triangular pyramid have?

     Match each pair of edges of the cube with an item in the box.
     (Lesson 10-1)                                                            A        B
                                     a. parallel edges
     24. AD and CG                                                                 D
                                                                                           C
                                     b. skew edges
     25. BC and EH                                                            E        F
                                     c. perpendicular
     26. CG and GH
                                        edges
                                                                                   H       G




                           Career – Urban Planner
     Workplace Knowhow



     A    n urban planner designs and supervises the construction of roads, buildings,
          tunnels, bridges, and water supply systems. The city of Berlin, Germany,
     changed rapidly following the removal of the Berlin Wall. New construction
     was needed to reconstruct the city and bring its buildings to modern standards.
     An urban planner in Berlin must decide what types of buildings are most
     appropriate for the needs of the city while maintaining the
     historical look and structure. Suppose an urban planner is
     beginning the design process for a new government building
     in Berlin.
     1. The shape of this new building is a right hexagonal prism.
        Make a sketch of the new structure, drawing the regular
        hexagons at the base and the top of the building.
     2. Draw a net (to scale) of the building. The sides of the regular
        hexagonal bases should have a length of 30 m, and the
        height of the building should be 60 m.
     3. What is the total surface area of the walls?
     4. How many marble tiles will be required to cover the outside
        walls if the tiles are 60 cm wide and 10 cm tall?

      mathmatters2.com/mathworks                       Chapter 10 Review and Practice Your Skills   431
10-3                                    Surface Area of Three-
                                        Dimensional Figures
                                        Goals
                                        Applications
                                                       I Find the surface area of three-dimensional figures.
                                                       Geography, Food service, Packaging, History


        Use a box such as a cereal box or a shoe box for the activity.
        1. Carefully tear the box apart at the edges. Do not keep the
           tabs that were used to glue the faces of the box together.
        2. How many polygons do you now have? Describe them.
        3. Use a ruler to measure. Then calculate the area of each
           of the polygons.
        4. Add the area of all the polygons. What does the total area
           represent?




      BUILD UNDERSTANDING
        In the previous lesson, you learned that the surface area of a polyhedron is
        the sum of the areas of all its bases and faces. You saw how the net of a three-
        dimensional figure can be used to find the total surface area. In this lesson,
        you will use formulas to find the surface area of three-dimensional figures.



Example 1
                                                                                          A
                                                                                                       4 in.
                                                                                              B
        Find the surface area of the rectangular prism.                            C
                                                                                              10 in.
                                                                                  6 in.
 Solution
        A rectangular prism has three pairs of congruent faces. The surface area is the sum
        of the areas of all the faces. Use the formula SA 2(area A) 2(area B) 2(area C),
        where each area can be found using the formula for the area of a rectangle, A lw.
        area of A      A   10 6
                                                              A           6 in.
                       A   60
                                                              10 in.
        area of B      A   10 4
                       A   40
                                                              B           4 in.                             Check
                                                              10 in.                                     Understanding
        area of C      A   6 4                            C       4 in.
                                                                                                       How is using the surface
                       A   24                             6 in.
                                                                                                       area formula similar to
                                                                                                       using nets to find the
             SA     2 (area A)     2 (area B)      2 (area C)                                          surface area of a three-
             SA     2 60    2 40        2 24                                                           dimensional figure?

             SA     248
        The surface area is 248 in.2.

       Chapter 10 Three-Dimensional Geometry
432
     A square pyramid with four congruent triangular faces is called a regular square
     pyramid. If you know the dimensions of just one triangular face, you can calculate
     the surface area of the pyramid.


Example 2
     GEOGRAPHY The Great Pyramid at Giza in Egypt has a
     square base approximately 756 ft long. The height of each
     triangular face, called the slant height, is approximately
     612 ft. What is the approximate surface area of the faces
     of the Great Pyramid?

 Solution
     The Great Pyramid is a regular square pyramid, so its faces
     are four congruent triangles. Calculate the surface area of
     one of the faces and multiply by 4.
                      1
              A         bh Use the formula for the area of a triangle.
                      2
                      1
              A           756 612 231,336
                      2
            SA     4 231,336          925,344
     The surface area of the faces of the Great Pyramid at Giza is approximately 925,344 ft2.

     A cylinder is a figure with two congruent circular bases and a curved surface. To
     find the surface area, add the areas of the bases to the area of the curved surface.

      Surface Area      SA     2 rh        2 r2
      of a Cylinder     where r is the radius of a base and h is the height of the cylinder.


Example 3                                                                         10 cm

     Find the surface area of the cylinder.

                                                                          25 cm
 Solution
              SA       2 rh        2 r2
              SA       2 3.14 10 25              2 3.14 102              3.14

              SA       1570     628       2198
     The surface area of the cylinder is approximately 2198 cm2.

     A cone is a three-dimensional figure with a curved surface and one
     circular base. The height of a cone is the length of a perpendicular                      slant
                                                                                            height        height
     segment from its vertex to its base. The slant height (s) of a cone is the                (s)
     length of a segment from its vertex to its base along the side of the cone.

        Surface       SA      rs      r2
       Area of a      where r is the radius of the base and s is the slant                         r
         Cone         height.


      mathmatters2.com/extra_examples             Lesson 10-3 Surface Area of Three-Dimensional Figures    433
Example 4
                                                                                     10 cm
        Find the surface area of the cone.

 Solution
        SA        rs            r2
                                                                                                      30 cm
        SA      3.14 10 30                    3.14 102                 3.14

        SA      942            314
        SA      1256
        The surface area of the cone is approximately 1256 cm2.



      TRY THESE EXERCISES
        Find the surface area of each figure. Round to the nearest hundredth.
          1.                                        2.                        17 m           3.           9 ft                    4.      2.8 m
                                                                                      10 m
                                                                                                                                                       4.3 m
               8 ft                                                                                              22 ft
                                      17 ft                                  42 m
                       6 ft                              21 m
                                                                     8m



          5.                                        6.           24 m                        7.                    0.4 cm         8.   7.2 mm

                                                                                    10 m                                 1 cm
               25 ft                                         26 m
                                        22.5 ft                               10 m
                                                                                                                                                       20 mm
                       22.5 ft                                                                               3.6 cm




      PRACTICE EXERCISES                            • For Extra Practice, see page 617.
        Find the surface area of each figure. Round to the nearest hundredth.
          9.                                  10.    9.2 cm
                                                                                       11.                                  12.         12 m
               1.8 m                                                                                                               5m
                                                                                             0.9 m
                                                                                                                   1.1 m
                                                                       26 cm
                                                                                                     1.1 m
                                4.5 m                                                                                                     30 m
                                                                                                                                                          13 m

        13.                                   14.                                      15.                                  16.
                                                                            20 ft                 10 m
                                                                                                                                   14.5 in.
                                                                                                     4m                                                  19 in.
                                  13 m                              22 ft                                                                     19 in.
                                                         24 ft
                                                                                                  10 m

                              6.6 m

        17. The base of each triangular face of a regular square pyramid is 9 in., and the
            height of each triangular face is 11 in. Find the surface area of the pyramid.

434     Chapter 10 Three-Dimensional Geometry
  18. PACKAGING Tamara is wrapping a gift. The box is 14 in. by 9 in. by 8 in.
      What is the minimum amount of wrapping paper she will need to cover the box?
  19. WRITING MATH Explain why the word “minimum” is used in Exercise 18.
  20. FOOD SERVICE Find the surface area of a can of cranberry sauce that has a
      height of 4.2 in. and a radius of 1.5 in.
  21. SPREADSHEET Use a spreadsheet to calculate the surface area of cylinders.
      Enter the formula for the surface area of a cylinder into the first cell. Then
      evaluate the formula for a cylinder that is 12 cm long with a radius of 3 cm,
      6 cm, 12 cm, and 24 cm.
  22. HISTORY An ancient Mayan pyramid has a square base. Each side of the
      base measures 230 m, and the slant height is 179 m tall. Find the area of the
      faces of the Mayan pyramid.



EXTENDED PRACTICE EXERCISES
  Find the surface area of each figure.
  23.   4m                            24.                  6m                       25.
                                                                                          10 m
                                                               8m
        4m                                                                                              12 m
                                            7m
                                                        12 m                                     12 m
             4m




  26. CRITICAL THINKING A cone has a height h and a radius r. Find an
      expression for the slant height. Then write a formula for the surface
      area of the cone using this expression.
  27. A hole is drilled through a solid cube that has edges of 4 cm. The hole is
      drilled perpendicular to the top face of the cube. The diameter of the hole
      is 2 cm. To the nearest whole number, what is the total surface area of the
      resulting solid figure?
  28. DATA FILE Refer to the data on U.S. shopping centers on page 570. What
      are possible dimensions of a rectangular prism whose surface area is equal
      to the area of Tysons Corner Center?



MIXED REVIEW EXERCISES
  Write the equation of each line using the given information. (Lesson 6-3)
  29. m      3, (1, 4)                                   30. m      2, ( 2,        3)
  31. (2, 7), (3, 9)                                     32. (0,    2), (2,    4)

  Find each product. (Lesson 9-5)
  33. (r     3) (r     4)                                34. (z     1) (z     4)
                  2
  35. (b     4)                                          36. (f     2) (f     5)
  37. (v     4) (v     8)                                38. (w     1) (w      7)
                                                                       2
  39. (3k     1) (2k        4)                           40. (2c     3)

   mathmatters2.com/self_check_quiz         Lesson 10-3 Surface Area of Three-Dimensional Figures              435
10-4                                 Perspective
                                     Drawings
                                     Goals           I Make one- and two-point perspective drawings.
                                                     I Locate the vanishing points of perspective drawings.
                                     Applications    Art, Interior design, History, Architecture


        Use the three photographs for Questions 1–3.
        1. Can you see depth in each of the photographs?
        2. Do you think the photographer was located above, below
           or at eye level with the subject?
        3. Does your eye travel to one point or two points in the
           background of each photograph?



      BUILD UNDERSTANDING
        A perspective drawing is a way of drawing objects on a flat
        surface so that they look the same as they appear in real life. The
        eye and the camera are both constructed so that objects appear
        progressively smaller the farther away they are. Parallel lines
        drawn in perspective appear to come together in the distance.
        Perspective drawings use vanishing points. A vanishing point is
        a point that lies on the horizon line. The horizon line is a line
        in the distance where parallel lines appear to come together.
        Perspective drawings can be made in either one-point
        perspective or two-point perspective.



Example 1
        Draw a cube in one-point perspective.

 Solution
        Step 1 Draw a square to show the front surface of the cube. Draw a horizon line j
               and a vanishing point A on line j.
        Step 2 Connect the vertices of the square to the vanishing point.
                     A               A                A
                            j               j                j




                   Step 1          Step 2           Step 3         Step 4

        Step 3 Draw a smaller square whose vertices touch the four line segments.
        Step 4 Connect the vertices of the two squares. Use a dashed segment to indicate
               the edges of the cube hidden from view. Remove line j and point A.

436    Chapter 10 Three-Dimensional Geometry
     A two-point perspective drawing has two vanishing points. If you are
     looking at the corner of an object in a perspective drawing, then it           Math: Who,
     is probably a two-point perspective drawing.                                   Where, When
                                                                                 Ancient Greeks and
                                                                                 Romans developed
                                                                                 techniques for drawing
Example 2                                                                        objects in perspective.
                                                                                 Renaissance artists in the
                                                                                 early 1400s used
     ARCHITECTURE Draw a building shaped like a rectangular prism                perspective drawings to
     in two-point perspective.                                                   create lifelike works.


 Solution
     Step 1 Use a straightedge to draw a vertical line       VP                VP
            segment to represent the height of the
            building. Draw a vanishing point (VP) on
            each side of the segment. Sketch depth
            lines from the top and bottom of the
            segment to each point.
     Step 2 Draw two segments parallel to the original       VP                VP
            segment to represent the height. Draw two
            segments to complete the top of the
            building.
     Step 3 Draw two more depth lines from the top
            corners to the vanishing points.
     Step 4 Erase the unused portions of the depth
            lines to complete the two-point
            perspective drawing.



Example 3
     Locate the vanishing point of
     the perspective drawing.

 Solution
     To locate the vanishing point, use a straightedge to draw the
     depth lines from the top edges of the figure. The point of their
     intersection is the vanishing point of the perspective drawing.




  TRY THESE EXERCISES
     Sketch each object in one-point perspective.
     1. cube          2. triangular prism        3. rectangular prism        4. cylinder
     5. your first name written in capital block-style letters

     Sketch each object in two-point perspective.
     6. cube                             7. shoe box                         8. table

      mathmatters2.com/extra_examples                            Lesson 10-4 Perspective Drawings        437
        Trace each perspective drawing to locate the vanishing point(s).
          9.                               10.                                 11.




      PRACTICE EXERCISES          • For Extra Practice, see page 617.

        Sketch each object in one-point perspective.
        12. window                         13. wall of a room                  14. couch

        Sketch each object in two-point perspective.
        15. television                     16. street sign                     17. school bus

        Trace each perspective drawing to locate the vanishing point(s).
        18.                                  19.                         20.




        21. WRITING MATH Describe the location of the
            vanishing point when a viewer can see three
            sides of a box drawn in one-point perspective.
        22. HISTORY The Sears Tower in Chicago is the
            tallest building in the U.S. Use a picture of the
            Sears Tower to make the one-point perspective
            drawing.
        23. INTERIOR DESIGN Make a one-point
            perspective drawing of the square tile pattern
            of a floor.

        A square is cut out of cardboard, and a flashlight shines behind it.
        Decide if it is possible to obtain the indicated shadow.
                                           24. square                          25. segment
                                           26. triangle                        27. point
                                           28. trapezoid                       29. kite
                                           30. pentagon                        31. parallelogram




438     Chapter 10 Three-Dimensional Geometry
  32. ARCHITECTURE Describe the perspective used
      by the architect to create the illusion of a three-
      dimensional house in her plans.
  33. Trace the house on a piece of paper. Locate a
      vanishing point.
  34. GEOMETRY SOFTWARE Use geometry software to
      make a one- or two-point perspective drawing. Begin
      by drawing a figure such as a rectangle and locating
      one or two vanishing points. Sketch the depth lines,
      and use them to complete the perspective drawing.


EXTENDED PRACTICE EXERCISES
  Determine the number of vanishing points used to draw each cube.
  35.                                  36.                                 37.




  38. RESEARCH Use your school’s library or another resource to find an example
      of a painting that uses one- and two-point perspectives. Identify the
      vanishing points used by the artists.
  39. CRITICAL THINKING Make a two-point perspective drawing of a rectangular
      box with a square window cut out of one of the faces. Show that the box has
      depth or thickness.
  40. CHAPTER INVESTIGATION Draw a line segment about 10 in. long across
      the center of a piece of paper with an arrowhead at each end. Place on the
      timeline in chronological order the name and accomplishment of each
      mathematician you researched. Label appropriately. Be sure the distances
      between the years recorded on your timeline are in proportion.


MIXED REVIEW EXERCISES
  State if it is possible to have a triangle with sides of the given lengths. (Lesson 5-4)
  41. 8, 5, 4                  42. 10, 6, 4       43. 12, 7, 6                   44. 9, 5, 5
  45. 16, 14, 12               46. 11, 8, 5       47. 7, 7, 2                    48. 12, 6, 5

  Factor each polynomial. (Lesson 9-7)
  49. 15z 2      18z                              50. 35r 2       60r
           5          4                                     3
  51. 8y        24y       28                      52. 6r s       9r 3s 2     18r 2s 2
  53. 28b 3c      84b 2c 5                        54. 24wx 3        12w 2x 2       20w 3x 3
  55. 24g 4h      40g 2h 2     16g 5              56. 30a 3b 2       45a 2b 3      60a 3b 3

   mathmatters2.com/self_check_quiz                             Lesson 10-4 Perspective Drawings   439
Review and Practice Your Skills
PRACTICE              LESSON 10-3
      Find the surface area of each figure.
        1.              8m                2.                  1.2 cm         3.         2 mm

                               6m                                 5 cm
               10 m
                                                                                             12 mm
                          6m

                                                          13 cm




        4.   6 cm                         5.                                 6.         4m
                                                                                   3m

                          cm                         7m
                       20                                     4m
                                                                                        10 m
                                                  4m
                                                                                                     5m


        7. A regular square pyramid has a base area of 100 cm2 and a height of 12 cm.
           What is the surface area? (Hint: Find the slant height s.)
        8. A can of peaches has a height of 5 in. and a diameter of 3 in. How much
           metal is needed to make the can?



PRACTICE              LESSON 10-4
      Sketch each object in one-point perspective.
        9. railroad track                10. house                           11. door

      Sketch each object in two-point perspective.
      12. stereo speaker                 13. grandfather clock               14. building

      Trace each perspective drawing to locate the vanishing point(s).
      15.                                      16.                           17.
                                                              1s
                                                                  tS
                                                                         t




      18. Explain the difference between a one-point perspective drawing and a
          two-point perspective drawing.

440   Chapter 10 Three-Dimensional Geometry
PRACTICE                 LESSON 10-1–LESSON 10-4
     Draw a net for each three-dimensional figure. (Lesson 10-2)
     19.                                          20.                                     21.




     Refer to the figure. (Lesson 10-1)
     22. How many vertices does the figure have?
     23. How many faces does the figure have?
     24. Name the figure.


     Mid-Chapter Quiz
     Identify each figure and name its base(s). (Lesson 10-1)
      1.                                T          2.           F                          3.       I               J
                     Q
                                                                                                H
                                                                                                           K
                                 S
                                                                B                                    M
           P                                            A                                           M               N
                                              U                          C
                          R                                                                     L              O
                                                            E        D

     Identify the three-dimensional figure for each net. (Lesson 10-2)
      4.                                           5.                                      6.




     Find the surface area of each figure. (Lesson 10-3)
      7.                                           8.                                      9.

                                                                         7 cm
                                      6 in.
           7 1 in.                                                                                      15 m
             2                                                      5 cm                                           12 m
                              8 in.
                                                                                                    12 m


     Sketch each object in one-point perspective. (Lesson 10-4)
     10. pentagonal prism                         11. building                            12. highway

                                                                    Chapter 10 Review and Practice Your Skills            441
10-5                                Isometric
                                    Drawings
                                    Goals          I Visualize and represent objects with isometric
                                                       drawings.
                                    Applications   Art, Recreation, History, Architecture


        Work with a partner. Use several cubes, preferably the kind that attach to each other.
        1. Join or stack the cubes together to build a structure. Hold the structure in your hand,
           and rotate it to obtain different views.
        2. Describe the structure to your partner. Which single view is easiest to describe?
        3. Make a new structure. Describe it to your partner so that it is clear how to build the
           same structure.
        4. Have your partner build the new structure based on your description.


      BUILD UNDERSTANDING
        In the last lesson, you learned how to reproduce three-dimensional
        figures on paper using perspective drawings. Another way to show
        a three-dimensional object is with an isometric drawing. An
        isometric drawing shows an object from a corner view so that three
        sides of the object can be seen in a single drawing.
        The figure shows an isometric drawing of a structure. In an
        isometric drawing, all parallel edges of the structure are shown as
        parallel line segments in the drawing. Perpendicular line segments
        do not necessarily appear perpendicular.

Example 1
                                                                                     Problem Solving
        Make an isometric drawing of a cube.                                               Tip
                                                                                    The length of all edges
                                                                                    in an isometric drawing
 Solution                                                                           are either the true
                                                                                    length or a scaled length.
        Step 1 Begin by drawing a vertical line, which will be the front edge       Angle measurements are
               of the cube. Then draw the right and left edges of the cube.         not preserved in
                                                                                    isometric drawings.




        Step 2 Draw the vertical sides of the cube parallel to the front edge.
               Complete the isometric drawing by sketching the rear edges of the cube.




442    Chapter 10 Three-Dimensional Geometry
Example 2
     Make an isometric drawing of a figure containing three cubes that form an
     L shape.

 Solution
     Step 1 Begin by drawing the left edges of the figure.
            Then draw the segments for the right edges of
            the figure. These segments are parallel to the
            edges of the left side and shown here in blue.
     Step 2 Complete the isometric drawing by sketching
            the five remaining segments that outline the cubes.


Example 3
     Use the isometric drawing to answer the questions.
     a. How many cubes are used in the drawing?
     b. How many cube faces are in the figure?
     c. If each cube face represents 3.5 ft2, what is the total
        surface area of the figure?

 Solution
     a. Even though there are only three visible cubes, the figure contains four cubes.
        The top cube is resting on a cube beneath it.
     b. To count the total number of cube faces, proceed in a logical manner so that
        you do not miss any sides.
            Cube faces pointing up          3
            Cube faces pointing to the front          6
            Cube faces pointing to the back           6
            Cube faces on the bottom            3
        There are a total of 3   6      6       3   18 cube faces on the figure.
     c. Since there are 18 cube faces and each face represents 3.5 ft2, the total surface
        area of the figure is 18 3.5 ft2 63 ft2.



  TRY THESE EXERCISES
     Create an isometric drawing of each figure.
     1. cube                                                  2. rectangular prism
     3. triangular prism                                      4. staircase composed of cubes

     5. ARCHITECTURE Using isometric grid paper, design a building using a
        rectangular prism for the base and a triangular prism for the roof.
     6. Use isometric grid paper to draw three different rectangular solids using a
        total of eight cubes each.

      mathmatters2.com/extra_examples                                 Lesson 10-5 Isometric Drawings   443
        Use the isometric drawing for Exercises 7–9.
          7. How many cubes are used in the drawing? How many are
             hidden?
          8. How many cube faces are exposed in the figure?
          9. If each face represents 1 yd2, what is the total surface area of
             the figure?



      PRACTICE EXERCISES           • For Extra Practice, see page 618.

        Create an isometric drawing of each figure.
        10. hexagonal prism                                 11. figure having 14 faces
        12. pentagonal prism                                13. figure composed of 11 cubes
        14. capital block-style letter M                    15. cube on top of a rectangular prism

        16. Use isometric grid paper to make a sketch of the letters of your first name.
        17. Use isometric grid paper to make a sketch of a rectangular prism with 18 faces.

        Use the isometric drawing for Exercises 18–20. Assume that
        no cubes are hidden from view.
        18. How many cubes are used in the drawing?
        19. How many cube faces are exposed in the figure?
        20. If the length of an edge of one of the cubes is 3.4 m, what is
            the total surface area of the figure?

        Give the number of cubes used to make each figure and the number of cube
        faces exposed. Be sure to count the cubes that are hidden from view.
        21.                             22.                           23.




        24.                             25.                           26.




        27. RECREATION Make an isometric drawing of a tent. Use a rectangular prism
            for the base and a triangular prism for the top of the tent.

        28. WRITING MATH Explain the similarities and differences between regular
            square grid paper and isometric dot paper. Which do you prefer to use for
            drawing?

444     Chapter 10 Three-Dimensional Geometry
  29. HISTORY Make an isometric drawing of the
      ancient Greek Parthenon.
  30. YOU MAKE THE CALL Ken says that it is
      impossible to make an isometric drawing of a
      rectangular prism containing 6 cubes that has
      exactly 24 cube faces showing. Brandon says
      that you can do this by making one long stack
      of cubes. Who is correct and why?



EXTENDED PRACTICE EXERCISES
  How many cubes are used to make each
  isometric figure? Be sure to count the cubes that        The Parthenon; Athens, Greece
  are hidden from view.
  31.                                 32.                                  33.




  34. ART Isometric grid paper can be used to make artistic patterns and designs
      similar to the ones shown. Use isometric grid paper to create two different
      artistic designs. Color the designs.




MIXED REVIEW EXERCISES
  Solve each system of equations graphically. (Lesson 8-2)
  35. 3x y 10                         36. 2x y    8                        37. x 2y 7
      x y 2                               x 3y        10                       4x 2y  22
  38. 2x    2y       8                39. 3x 2y       12                   40. x    3y     11
      2x    2y   0                        x 3y         7                       x    2y       7

  Show each sample space using ordered pairs. (Lesson 4-3)
  41. A quarter and a dime are tossed.
  42. A six-sided number cube is rolled and a dime is tossed.
  43. A six-sided number cube is rolled and a spinner with five different colors is
      spun. (Use the numbers 1–5 for the colors on the spinner.)
  44. Two spinners, one numbered 1–4 and the other lettered A–F, are spun.

   mathmatters2.com/self_check_quiz                            Lesson 10-5 Isometric Drawings    445
10-6                                 Orthogonal
                                     Drawings
                                     Goals          I Sketch orthogonal drawings of figures.
                                                    I Sketch and use foundation drawings.
                                     Applications   Engineering, Interior design, History, Safety


        Use 11 cubes, preferably interlocking ones.                                     D                           C

        1. Label the four corners of a sheet of paper A, B, C, and D.
        2. Build a structure by stacking the given number of cubes in                               3   2
           each position as shown in the figure.
                                                                                                    1   2    3
        3. Match each figure below with corner views A, B, C, or D.
           Try to visualize the answer in your mind; then check the                     A                           B
           actual structure to verify.




                    figure 1          figure 2          figure 3             figure 4




      BUILD UNDERSTANDING
        While perspective drawings and isometric drawings are useful to help visualize a
        three-dimensional figure, they may not show all the details of an object that you
        need. An orthogonal drawing, or orthographic drawing, gives top, front, and side views
        of a three-dimensional figure as seen from a “straight on” viewpoint. In an
        orthogonal drawing, solid lines represent any edges that show.




Example 1
        Make an orthogonal drawing of the figure.
                                                                                             top
 Solution
        Draw the front view first.
        Draw the top view above it. Make sure it has the same width as
        the front view.                                                                     front           right
                                                                                                             side
        Draw the right-side view. Make sure it has the same height as
        the front view and the same depth as the top view.

446    Chapter 10 Three-Dimensional Geometry
Example 2
     ENGINEERING The isometric drawing shows part of an air-
     conditioning duct.
     a. Make an orthogonal drawing of the duct showing the front, top,
        and right-side views.
     b. The length of one cube side is 1.6 m. The building engineer needs the surface
        area of the top to be between 10 m2 and 10.5 m2. Is the duct within the
        necessary requirements?

 Solution
     a. Think of the duct as a combination of four cubes. Then make
        each view of the orthogonal drawing.
                                                                                            top
     b. Calculate the surface area of the top of the figure. It is
        composed of four cube faces.
             1.6 1.6    2.56 m2         Calculate the area of one cube.                    front        right side
                                  2
             2.56 4    10.24 m          Find the total surface area of the top.

          The duct is within the given requirements.
                                                                                            54     67   54
     A foundation drawing shows the base of a structure and the height of
     each part. They are often used by architects and design engineers.
     The figure shows a foundation drawing of the Sears Tower in                            67     98   67
     Chicago. Each number represents how many stories are in each
     section.                                                                               41     98   41




Example 3
     Create a foundation drawing for the
     isometric drawing. Assume the drawing is
     viewed from the lower left-hand corner.

 Solution
     First draw the orthogonal top view of the figure.
                                                                           3      2   1
     Then determine how many cubes belong in each
     section, and write the number to complete the
                                                                           2
     foundation drawing.

                                                                           1

  TRY THESE EXERCISES
     Make an orthogonal drawing labeling the front, top, and right-side views.
     1.                      2.                            3.                         4.




      mathmatters2.com/extra_examples                                     Lesson 10-6 Orthogonal Drawings      447
        The figure is a cube with a “half cube” on top of it. Tell which view of the figure
        is shown.
          5.




          6.

          7.




        Create a foundation drawing for each figure. Assume the drawing is viewed
        from the lower left-hand corner.
          8.                                9.                                10.




      PRACTICE EXERCISES
      PRACTICE EXERCISES          • For Extra Practice, see page 618.
        Make an orthogonal drawing labeling the front, top, and right-side views.
        11.                                12.                                13.




        For each foundation drawing, sketch the front and right orthogonal views.
        14.                                15.                                16.
               5           5                      1     2      3                     1


               3           1                            2      1                     3        2

                   front                               front
                                                                                     3        2   1

                                                                                          front


        17. WRITING MATH Choose an object and describe how the orthogonal
            drawings of the front, top, and right-side views are different or the same.
        18. Make an orthogonal drawing of a regular six-sided number cube showing a
            view of each of the six sides.

448     Chapter 10 Three-Dimensional Geometry
  19. SAFETY Most skyscrapers are built using steel girders
      shaped like the capital letter I. The shape avoids buckling
      and can support heavy loads. Make an orthogonal drawing
      of the girder showing the front, top and right-side views.
  20. WRITING MATH Investigate the meaning of the word
      “orthogonal.” Why do you think it is used to describe the
      “straight on” view of a figure?
  21. HISTORY Make an orthogonal drawing of the Sears Tower
      in Chicago. (Refer to the foundation drawing of the Sears
      Tower found earlier in this lesson.)


EXTENDED PRACTICE EXERCISES
  Create an orthogonal drawing for each figure.
  22.                                 23.                                     24.




  25. CRITICAL THINKING The drawing shows top and front views of a
      building. Draw three possible right-side views of the building.
  26. Make an orthogonal drawing showing the front, top and right-side
      views of a staircase that is one cube wide and reaches a height of                        top
      ten cubes. Then make a foundation drawing of the staircase.
  27. INTERIOR DESIGN If the height of each cube in Exercise 26 is 8 in.,
      how many square inches of carpeting are needed to completely
      cover the front and top edges of the staircase?                                           front



MIXED REVIEW EXERCISES
  Use grid paper. Graph each function for the domain of real numbers. (Lesson 6-6)
  28. y    x2       1                 29. y   x2      x       2                30. y   x2       x       4
  31. y    2x 2     3x   1            32. y   3x 2        4                    33. y   2x 2         x   1
                2                                 2                                         2
  34. y    2x       2                 35. y   x       3x      4                36. y   2x           x   5

  Write each phrase as a variable expression. (Lesson 2-3)
  37. six times a number divided by three
  38. the product of seven and a number
  39. two more than 12 times a number
  40. the quotient of negative six and a number
  41. a number less nine
  42. negative three times a number

   mathmatters2.com/self_check_quiz                               Lesson 10-6 Orthogonal Drawings           449
Review and Practice Your Skills
PRACTICE             LESSON 10-5
      Use the isometric drawing for Exercises 1–3. Assume that no
      cubes are hidden from view.
        1. How many cubes are used in the drawing?
        2. How many cube faces are exposed in the figure?
        3. If the length of one of the cubes is 2.8 in., what is the total
           surface area of the figure to the nearest tenth of an inch?

      Create an isometric drawing of each figure.
        4. pentagonal prism                                  5. figure composed of 10 cubes
        6. figure with 7 faces                               7. triangular prism


PRACTICE             LESSON 10-6
      Create a foundation drawing for each isometric drawing. Assume the drawing
      is viewed from the lower left-hand corner.
        8.                                     9.                              10.




      Make an orthogonal drawing labeling the front, top and right-side views.
      11.                                12.                                 13.




      For each foundation drawing, sketch the front and right orthogonal views.
      14.                                   15.                                16.
             3       2                              3   3                                    1

             2       1                              2   2                            1       2

                 front
                                                    1   1                                    1

                                                    front                                front

450   Chapter 10 Three-Dimensional Geometry
PRACTICE        LESSON 10-1–LESSON 10-6
     Draw a rectangular pyramid. (Lesson 10-1)
     17. How many vertices does it have?
     18. How many faces does it have?
     19. How many bases does it have?

     20. What is the surface area of a rectangular prism with dimensions of
         2 in. by 6 in. by 4 in.? (Lesson 10-3)
     21. Draw a net for a pentagonal prism. (Lesson 10-2)

     For each foundation drawing, sketch the front and right orthogonal views. (Lesson 10-6)
     22.                                23.                                       24.
           3    3      1                      2       1                                 1      2     1
                                                             EER   FEATURE #2
           2                                  2       1                                        1

                                                  front                                      front
           1

               front       EXHIBIT DESIGNER


                            Career – Exhibit Designer
     Workplace Knowhow
                                                                              Empire State Building

     E    xhibit designers organize and design products
          and materials so they serve the intended
     purpose and are visually pleasing. Some exhibit
                                                                    Section
                                                                               Width*
                                                                               (feet)
                                                                                            Depth*
                                                                                            (feet)
                                                                                                         Height*
                                                                                                          (feet)
     designers work in museums to create, design,                      1         833         375          125
     install, disassemble and store historical displays.               2         667         292          250
     A museum exhibit designer is designing models                     3         500         208          750
     of notable skyscrapers for a historical exhibit.                  4         250         125          125
     Accurately communicating the design ideas to                  * Dimensions are not actual measurements
     the model creator is important. The figure shown                    of the Empire State Building
     is a two-point perspective drawing of the Empire
     State Building (without the lightning rod) in New                                                    4
     York City, NY.
     Refer to the drawing and the table.                                                                  3
                                                                                        height
     1. Create an orthogonal drawing of the Empire State Building.
     2. Create an isometric drawing of the Empire State Building.                                         2
     3. Create a one-point perspective drawing of the Empire State Building.                                  1
                                                                                                               width
                                                                                             depth



      mathmatters2.com/mathworks                          Chapter 10 Review and Practice Your Skills               451
10-7                                        Volume of Prisms
                                            and Pyramids
                                            Goals           I Use a formula to find the volume of prisms.
                                                            I Use a formula to find the volume of pyramids.
                                            Applications    Earth science, Hobbies, Packaging, History, Machinery


        Use sheets of heavy paper, a ruler, tape and dried beans.
                                                                                             10 cm
        1. Use the patterns to make a cube and a square pyramid. Both
           shapes should be missing one side.
        2. Completely fill the cube with dried beans. Record the
           number of beans used.
        3. Pour the beans directly from the cube to the pyramid.
           Approximately how many times can you fill the pyramid?
        4. What relationship do you see?                                                                        10 cm

                                                                                                       10 cm


      BUILD UNDERSTANDING
        Recall that volume is a measure of the number of cubic units needed to fill a
        region of space. To find the volume of any prism, multiply the area of the base
        by the height of the prism.


             Volume of          V   Bh
              a Prism           where B is the area of the base and h is the height of the prism.


Example 1
        Find the volume of each prism.                                                           Problem Solving
                                                                                                       Tip
        a.                                            b.
                          25 m                                      7 ft
                                                                                                Volume is always
                                                                                                measured in cubic units
                                                                   7 ft                         since it is found by
                                                            7 ft
                         12 m
              14 m                                                                              multiplying an area
                                                                                                (which is measured in
                                                                                                square units) by a length.
 Solution
        a. The base is a rectangle.                   b. The base is a square.
                     B    14 12            B   lw              B    72              B   s2

                     B    168                                  B    49
             Use the volume formula.                       Use the volume formula.
                     V    168 25           V   Bh              V    49 7            V   Bh

                     V    4200                                 V    343
             The volume is 4200 m3.                        The volume is 343 ft3.

452    Chapter 10 Three-Dimensional Geometry
Example 2                                                               14 cm       9 cm

     Find the volume of the triangular prism.
                                                                                25 cm

 Solution
     The base is a right triangle.
                   1                            1
            B           9 14              B       bh                                        Problem Solving
                   2                            2
            B      63                                                                             Tip
     Use the volume formula.                                                               To find the volume of a
                                                                                           rectangular prism, use
            V      63 25                  V    Bh                                          the formula V lwh.
            V      1575                                                                    To find the volume of a
                                                                                           cube, use the formula
     The volume is 1575 cm3.                                                               V s 3.



Example 3
     RECREATION A rectangular swimming pool is filled with 1750 ft3 of water. The
     pool width is 14 ft and the pool depth is 5 ft. Find the length of the pool.

 Solution
     Use the volume formula for a rectangular prism.
                1750    l 14 5                 V    lwh

                1750    70l                    Solve for l.
                1750     70l
                 70      70
                  25    l
     The length of the pool is 25 ft.

     The volume of a prism is 3 times the volume of a pyramid. To find the volume of
     a pyramid, take 1 the product of the area of the base and the height.
                         3

                               1
       Volume of        V        Bh
                               3
       a Pyramid        where B is the area of the base and h is the height.



Example 4                                                                                      30 m

     Find the volume of the rectangular pyramid.
                                                                                                              16 m
                                                                                                      24 m
 Solution
     Find the area of the rectangle base.                     Use the volume formula.
                                                                         1                     1
            B     24 16               B   lw                        V        384 30        V     Bh
                                                                         3                     3
            B     384 m2                                            V   3840
                                                          3
     The volume of the pyramid is 3840 m .

      mathmatters2.com/extra_examples                         Lesson 10-7 Volume of Prisms and Pyramids          453
      TRY THESE EXERCISES
        Find the volume of each figure.
          1.            8m           2.               9 mm             3.                          4.                     6 in.
                                           10.3 mm
                                                                                                                              12 in.
                                                                             21 m
                              23 m         18 mm                                                         8 in.
                                                                                                                             6 in.
                                                                                                                 12 in.
                10 m                                                                        11 m
                                                                                    14 m



          5. PACKAGING A gift box that is 15 in. high is a prism with a square base. It
             has a volume of 540 in.3. What is the length of the sides of its base?
          6. Find the volume of a square pyramid if the length of a side of its base is 23 ft
             and its height is 65 ft. Round the answer to the nearest tenth.
          7. A triangular prism has a volume of 54 cm3. The triangular base has a height of
             4 cm and a base of 6 cm. What is the height of the prism?
          8. WRITING MATH Given the volume of a pyramid and the area of its base,
             describe how you can find its height.



      PRACTICE EXERCISES
      PRACTICE EXERCISES                   • For Extra Practice, see page 619.
        Find the volume of each figure.
          9.                         10.                     7m        11.                         12.                4m 4m
                                                                                                           3m
                                            6m
                                                          9.5 m                                            8m
               24.4 m
                                                                                                                                  10 m
                                                                               31 cm                                12 m
                              9m
                        9m



        13.                          14.                               15.                         16.              15 in.
               17 ft         18 ft                                                                        7.2 in.

                                            14 m                                           20 m
                                                                                                                  12 in.
                22 ft                                                         10 m
                                                                  6m
                                                     8m                                    6m
                                                                                     6m

        17. WRITING MATH Explain the difference between the surface area
            and the volume of a three-dimensional figure.
        18. A cereal box is 8 in. long and 4 in. wide. Its volume is 368 in.3. What
            is the height of the box?
        19. The perimeter of the base of a square prism measures 60 cm. The
            height of the prism is 25 cm. What is the volume of the prism?
        20. The base of a prism is an isosceles right triangle with sides
            measuring 5 2 cm, 5 2 cm and 10 cm. The height of the prism
            is 20 cm. What is the volume of the prism?

454     Chapter 10 Three-Dimensional Geometry
  21. MACHINERY A dump truck has a bed that is 12 ft long and 8 ft wide. The
      walls of the bed are 4.5 ft high. When the truck is loaded, no material can be
      higher than the walls of the bed. If topsoil costs $18/yd3, what is the cost of a
      full truckload of topsoil?
  22. EARTH SCIENCE When water freezes, its volume increases by about 10%. A
      tin container that measures 1 ft by 10 in. by 9 in. is exactly half full of water. It
      is left outside on a winter day, and all of the water freezes. What is the
      approximate volume of ice in the tin?
  23. HISTORY The square base of an Egyptian pyramid is 62 m long. If the height
      of the pyramid is 78 m, what is its volume?


EXTENDED PRACTICE EXERCISES
                                                                                                                           2m
  Find the volume of each figure.
  24.                                           25.                         17 cm                26.                                   2m

                                                      4 cm
        5 ft          5 ft                                                                              10 m                    20 m
                                                                              26 cm
                                  7 ft
                                                             21 cm
               6 ft                                                                                                 10 m


  27. HOBBIES An aquarium has a length of 15 in. and a width of 11 in. A rock put
      into the aquarium causes the water level to rise by 2 in. The rock is
      completely submerged. What is the volume of the rock?
  28. Suppose you wish to make a cardboard box with a volume of 1000 cm3. What
      dimensions would you give to the box in order to use the least amount of
      cardboard?
  29. CRITICAL THINKING The sides of a cube each measure 1 ft. If each side is
      increased by 1 in., by how many cubic inches would the volume increase?
  30. CHAPTER INVESTIGATION Research a significant historical event that was
      happening at each date on your timeline.


MIXED REVIEW EXERCISES
  Simplify. (Lesson 9-4)
  31. 2z(4z           w)                                             32. 3d(2d       3c)
  33. 5rs(3r          4s)                                            34.   6xy(2x        5y)
  35. 2p(4p           3r)        5p(7p    8r)                        36. 3st(3s      3t)         2st(2s            5t)
  37. 4yz(y           3z)        2yz(6y   4z)                        38.   5cd(2c        3d)          2cd(c          5d)

  Copy quadrilateral KLMN on grid paper. Draw                                            4
                                                                                             y

  each dilation. (Lesson 7-5)                                                    K                     L
                                                                                         2
  39. scale factor 2, center (0, 0)
                                                                                                               x
                             1
  40. scale factor , center (0, 0)                                           4       2            2        4
                             2
  41. scale factor 3, center L                                                           2
                             2                                                   N                     M
  42. scale factor , center N                                                            4
                             3


   mathmatters2.com/self_check_quiz                           Lesson 10-7 Volume of Prisms and Pyramids                           455
10-8                                          Volume of Cylinders,
                                              Cones and Spheres
                                              Goals
                                              Applications
                                                               I Find the volume of cylinders, cones and spheres.
                                                               Sports, Horticulture, Astronomy, History, Chemistry


        FITNESS A person’s vital capacity is the measure of the
        volume of air held in his or her lungs. Take a deep
        breath, and blow into a balloon as much air as
        possible. Trap the air by tying off the balloon.
        1. Push on the end of the balloon so that it forms a
           sphere. Then use a tape measure to find the
           circumference of the balloon. Let this measure be C.
        2. Find your vital capacity (VC) by using the formula
                 C3
           VC      2.
                   6




      BUILD UNDERSTANDING
        Recall that the general formula for finding the volume of a prism is V Bh,                            r
        where B is the area of the base. Since the base of a cylinder is circular, replace
        B in this formula with r 2, the formula for the area of a circle.
                                                                                                                       h
                                     2
            Volume of    V          r h
            a Cylinder   where r is the radius of the base and h is the cylinder’s height.

                                                                  4.5 m


Example 1
                                                                                                    Check
                                                                              10 m
        Find the volume of the cylinder.                                                         Understanding
                                                                                              How is the formula for
 Solution                                                                                     the volume of a cone
                                                                                              similar to the formula for
        V       r 2h                     Use the formula for the volume of a cylinder.        the volume of a
                                                                                              pyramid?
                         2
        V     3.14 (4.5)       10            3.14

        V     635.85
        The volume of the cylinder is approximately 635.85 m3.

        The volume of a cylinder is 3 times the volume of a cone that has the same
                                                                                                              h
        radius and height. To find the volume of a cone, take 1 the product of the
                                                              3
        area of the base and the height of the cone.                                                              r

                                1
            Volume of      V      r 2h
                                3
             a Cone        where r is the radius of the base and h is the cone’s height.


456    Chapter 10 Three-Dimensional Geometry
Example 2                                                                            4 cm

     Find the volume of the cone.
                                                                                            6 cm

 Solution
            1 2
     V        r h                   Use the formula for the volume of a cone.
            3
            1
     V         3.14    42 6             3.14
            3
     V     100.48
     The volume of the cone is approximately 100.48 cm3.

     There is also a formula for the volume of a sphere.

                               4
         Volume of        V      r3
                               3
         a Sphere         where r is the radius of the sphere.



Example 3
     ASTRONOMY The diameter of the planet Mars is
     approximately 6800 km. What is the volume of Mars?

 Solution
     Assume that Mars is a sphere. Mars’ radius is half of
     6800 km, or 3400 km.
            4 3
     V        r               Use the formula for the volume of a sphere.
            3
            4
     V         3.14    (3400)3           3.14
            3
     V     1.6455 1011 km3
     The volume of Mars is approximately 165,000,000,000 km3.                                              Mars

                                                                  5m
                                                                         10 m
Example 4
                                                         12 m
     Find the volume of the figure.

 Solution
     The total volume of the figure is the sum of the cylinder’s volume and the
     cone’s volume.
              Volume of the cylinder                     Volume of the cone
                      2                                         1 2
              V       r h                                V        r h
                                                                3
                                                                1
              V     3.14 122 5                           V         3.14     122 10
                                                                3
              V     2260.8                               V      1507.2
     Total volume          2260.8      1507.2     3768
     The volume of the figure is approximately 3768 m3.

      mathmatters2.com/extra_examples                 Lesson 10-8 Volume of Cylinders, Cones and Spheres   457
      TRY THESE EXERCISES
        Find the volume of each figure. Round to the nearest whole number.
          1.                             2.                    19 in.       3.          5 cm                     4.                  8 in.

                         24 m
               11.2 m
                                                                                                 10.4 cm

                                                                                                                                14 in.



          5. A cone with a radius of 12 cm has a volume of 753.6 cm3. What is the height
             of the cone?
          6. What is the volume of a sphere with a radius of 35 m?
          7. A cubic foot of sand has a mass of 75 lb. How many pounds of sand would fit
             into a cylindrical container with a radius of 10 ft and a height of 5 ft?
          8. What is the volume of a hemisphere with a radius r?



      PRACTICE EXERCISES               • For Extra Practice, see page 619.

        Find the volume of each figure. Round to the nearest whole number.
          9.              12 m   10.                 17 in.         11.                                      12.

                                        13 in.                                                  16 m
                                                                                                                                             10 m

                                                                                        9.2 m
                                                                                                                                      12 m
                                                                                                                      10 m


        13. A gasoline storage tank is a cylinder with a radius of 10 ft and a height of 6 ft.
            How many cubic feet of gasoline will the tank hold?
        14. A sphere has a volume of 523 1 in.3. What is the radius of the sphere?
                                                 3
        15. SPORTS Tennis balls are sold in cylindrical cans. Each can
            holds three tennis balls. If the volume of the can is 150.72 in.3,
            what is the approximate radius of a tennis ball?
        16. HORTICULTURE The figure at the right is a sketch of a
            proposed greenhouse that is to be shaped like a hemisphere. To                                                   50 ft
            the nearest cubic foot, what is the amount of space inside this
            greenhouse?

        Find the volume of each figure. Round to the nearest tenth.
        17.       7 cm
                                           18.          4 ft
                                                                                                           19.        10 cm
                                                                          4 ft

                                                 1 ft                            1 ft
                                                                                                                                     4 cm
                                                        4 ft            4 ft

                                                 The radius of the hole is 1 ft.                                      16 cm


        20. WRITING MATH Explain how to find the volume of a cylinder, a cone, and a
            sphere. Describe the similarities among the three different volume formulas.

458     Chapter 10 Three-Dimensional Geometry
  21. HISTORY Early astronomers estimated the equatorial
      radius of the earth to be approximately 520 km. The
      actual radius is about 6378 km. What is the
      approximate volume of the earth? How inaccurate was
      the astronomers’ estimate of the earth’s volume?
  22. A cylinder has a circumference of 12 m and a height
      of 4.7 m. What is the volume of the cylinder? Round the
      answer to the nearest whole number.
  23. CHEMISTRY A chemist pours 188.4 cm3 of a liquid
      into a glass cylinder that has a radius of 2 cm. How
      many centimeters deep is the liquid?

  DATA FILE For Exercises 24–27, refer to the data on
  various balls used in sports on page 571. Find the volume
  of each sports ball. Round to the nearest tenth.
  24. baseball                    25. golf ball                 26. volleyball              27. croquet ball

EXTENDED PRACTICE EXERCISES
  CRITICAL THINKING Each of the cans shown in the diagram has a volume of
  1000 cm3. (Recall that 1000 cm3 1 L.)
                                                    3 cm
                                        4 cm
    5 cm
                     6 cm
                                                                  7 cm
                                                     1L                            8 cm
                                         1L
        1L            1L                                              1L               1L
        a              b                  c          d                e                f


  28. Find the height of each can rounded to the nearest tenth.
  29. Which two cans have the least surface area?
  30. If you manufacture one liter cans, what radius would you use so that you
      save the most material?

  31. SPREADSHEET Write a spreadsheet program to calculate the volume of a
      sphere. Input the formula in the first cell, and calculate the volumes of
      spheres with radii 2 m, 4 m, 8 m and 16 m. Do you notice a pattern? Explain.

MIXED REVIEW EXERCISES
  Find the value of the determinant of each matrix. (Lesson 8-5)
       4 5                  1 8               6     2                              5 6                 0        1
  32.                 33.               34.                 35.                                  36.
       3 6                  2 5               4     7                             12 8                 2        6

             3   3                  8         13            4     7              4 9                       11       4
  37.        4              38.                     39.                    40.                   41.
                 6                 10          9            2     8              2 5         M              9       3
                                                                                                   N
  In the figure, m MLN 35° and m NLO                       55°.
  Find each measure. (Lesson 5-8)
                                                                                            L
       _
  42. mNO                                      -
                                         43. m MN                                                      O

        -
  44. mMNO                                      -
                                         45. m MPO                                 P


   mathmatters2.com/self_check_quiz                Lesson 10-8 Volume of Cylinders, Cones and Spheres               459
Review and Practice Your Skills
PRACTICE             LESSON 10-7
      Find the volume of each figure. Round to the nearest tenth.
        1.           4m                         2.               4 mm           3.
                                                        12 mm                         9m

                           15 m
                                                     8.4 mm                                            13 m
                                                                                        10 m
              5m


        4.                      2 in.           5.                              6.                            7m

                                                       13 m                          5m
                                        6 in.
                                                                                                         9m
             4 in.
                                   3 in.                         8m
                        7 in.                             6m

        7.                                      8.                              9.




                                                                                                5
                                                                                     12 cm




                                                                                                 cm
                                  6 cm
                                                       19 m

                                6 cm                                                           18 cm
                 6 cm
                                                                         11 m
                                                          11 m


      10. A triangular prism has a volume of 108 m3. The triangular base has a height
          of 9 m and a base of 6 m. What is the height of the prism?


PRACTICE             LESSON 10-8
      Find the volume of each figure. Round to the nearest whole number.
      11.                                        12.                                   13.                8 in.

                           24 m
             10 m                                                7 in.
                                                                                             14 in.



      14.                                        15.             5m                    16.      6 cm



                                                                         9m
                                11 m
                                                                                             10 cm




      17. The volume of a cone with a 14-mm diameter is 820 mm3. Find the height.
      18. The radius of an inflated beach ball is 15 in. What is the amount of air inside?

460   Chapter 10 Three-Dimensional Geometry
PRACTICE          LESSON 10-1–LESSON 10-8
     19. Find the minimum amount of gift wrap needed to cover a box that measures
         2 ft by 3 ft by 6 in. (Lesson 10-2)
     20. Sketch a net for the three-dimensional figure. (Lesson 10-2)
     21. Create an isometric drawing of a triangular prism.
         (Lesson 10-5)
     22. Create an isometric drawing of a figure composed of six cubes.
         (Lesson 10-5)

     Find the surface area and volume of each figure. Round to the nearest tenth.
     (Lessons 10-3, 10-7 and 10-8)
     23.                               24.               18 cm                  25.
                                                                                                                  10 m

                                                                        23 cm
                      16 in.
                                                                                      8m
                                                                                                           17 m
             12 in.                                                                           6m



     26.                               27.                                      28.
                                             7.02 ft          7.3 ft                  3 in.


                                                                 4 ft                      3 in.
           7 mm                                                                                    3 in.
                               22 mm

                  8 mm                                 4 ft




     Identify each figure and state the number of faces, vertices and edges.
     (Lesson 10-1)
     29.                               30.                                      31.




     Trace each perspective drawing to locate the vanishing point(s). (Lesson 10-4)
     32.                               33.                                      34.




                                                         Chapter 10 Review and Practice Your Skills               461
10-9                                 Problem Solving Skills:
                                     Length, Area, and Volume
      Formulas are mathematical tools that help guide you to a solution.
      The strategy, use an equation or formula, is appropriate when you have      Problem Solving
      data that can be substituted into a formula or equation. If you do not         Strategies
      know the formula, write an equation that states the relationships in          Guess and check
      the problem. In either case, solve the equation to find the solution.
                                                                                    Look for a pattern
                                                                                    Solve a simpler
                                                                                    problem

Problem                                                                             Make a table, chart
                                                                                    or list
      CONSTRUCTION The tank of a raised cylindrical water tower is 26 ft            Use a picture,
      high with a radius of 11 ft. There is a 2-ft-wide walkway around the          diagram or model
      base of the tank.                                                             Act it out

      a. About how many gallons of paint are needed to paint the exterior           Work backwards
         of the tank if 1 gal of paint will cover 300 ft2?                          Eliminate
                                                                                    possibilities
      b. There is a railing around the walkway. How long is it?
                                                                                  ✔ Use an equation or
      c. How many gallons of water does the tank hold? (The volume of               formula
         1 gal is approximately 0.1337 ft3.)


 Solve the Problem
      a. Each gallon of paint covers an area of 300 ft2.
            SA     2 r 2 2 rh   Use the formula for surface area of a cylinder.
                              2
            SA     (2 3.14 11 ) (2 3.14 11 26)                3.14
            SA     760 1796 2556
         The surface area is approximately 2556 ft2. Since each gallon
         of paint covers 300 ft2, divide the surface area by 300 ft.
            2556     300    8.5
         It will take approximately 8.5 gal of paint.
      b. The length of the railing is the circumference of a circle
         with a radius 2 ft greater than the radius of the tank.
            C    2 r   Use the formula for the circumference of a circle.
            C    2 3.14 (11 2)              3.14
            C    81.6
         The railing is approximately 81.6 ft long.
      c. The number of gallons the tank will hold is the volume of the tank.
            V      r 2h Use the formula for the volume of a cylinder.
                        2
            V    3.14 11 26 9878.4                3.14

         The volume is approximately 9878.4 ft3. Since 1 gal is approximately
         0.1337 ft3, the tank holds approximately 9878.4 0.1337 73,884.8 gal.

462   Chapter 10 Three-Dimensional Geometry
TRY THESE EXERCISES                                                                             Five-step
                                                                                                   Plan
  Round answers to the nearest tenth.
                                                                                                1   Read
   1. A cypress tree has a radius of 18.75 ft. If it is fenced in so that there is a            2   Plan
      border 8 ft wide around the trunk of the tree, how many feet of fencing                   3   Solve
                                                                                                4   Answer
      are needed?                                                                               5   Check
   2. South African ironwood is the world’s heaviest wood, weighing about
      90 lb/ft3. How much would an 8-in. cube of ironwood weigh?
   3. PACKAGING Find the minimum amount of paper needed for the label of a
      cylindrical soup can that is 7 in. high with a radius of 2.25 in.
   4. The windows of a building have a total area of 32.4 m2. Each
      window is a rectangle measuring 1.2 m by 1.8 m. How many
      windows are in the building?


PRACTICE EXERCISES
  Round answers to the nearest tenth.
   5. LANDSCAPING A pound of grass seed covers an area of
      250 ft2. How many pounds of seed would you need for a
      rectangular lawn measuring 35 ft by 50 ft?
   6. The roof of a shed is a square pyramid with sides of 2.4 m.
      The height of each triangular face is 2 m. How many square
      meters of tar paper would it take to cover the roof ?
   7. A circular swimming pool has a diameter of 17 ft and a depth
      of 5 ft. If the pool is considered to be full when the water level
      is 1 ft below the rim of the pool, how many cubic feet of water
      does it take to fill the pool?
   8. A cardboard hat is made of a cone with radius 5 in. and slant height 14 in.
      The circular brim is 4-in. wide. Find the minimum amount of cardboard
      needed to make the hat.
   9. WRITING MATH Write a problem that can be solved by applying the
      formula for length, area, or volume. Provide an answer for your problem.
  10. CRITICAL THINKING The names and sizes of wooden boards specify
      the dimensions before they are dried and planed. For example, a
      one-by-ten board is actually 3 in. by 9 1 in. If a patio is built using 15
                                   4          4
      one-by-ten boards that are each 12 ft long, what is the area of the patio?

MIXED REVIEW EXERCISES
  Simplify. (Lesson 9-3)
        14b 2               51a 2c 3                27g 3h              7m 3n 4              49k 3l 2
  11.                 12.                     13.                 14.                  15.
         2b                  3ac                     9gh                 m2n                 14k 3l 2

  Calculate each permutation. (Lesson 4-6)
  16. 8P2             17. 5P2                 18.   12P4          19.   10P6           20. 9P4
  21.   12P8          22. 9P6                 23.   10P3          24.   14P5           25.   11P3


                                       Lesson 10-9 Problem Solving Skills: Length, Area and Volume           463
       Chapter 10 Review
VOCABULARY
       Choose the word from the list that best completes each statement.
                  ?__
        1. A(n) ___ is a polyhedron with only one base.
                                                                                      a. cone
                  ?__
        2. A(n) ___ is a polyhedron with two identical parallel bases.
                                                                                      b. cylinder
                  ?__
        3. The ___ of a cone is the length of a segment drawn from its
                                                                                      c. face
           vertex to its base along the side of the cone.
                                                                                      d. isometric
                  ?__
        4. A(n) ___ is the set of all points in space that are a given
           distance from a given point.                                               e. orthogonal
        5. A one-point perspective drawing has one ___
                                                     ?__.                                 f. prism
        6. A(n) ___ is a three-dimensional figure with two congruent
                   ?__                                                                g. pyramid
           circular bases that lie in parallel planes.                                h. slant height
        7. The sum of the areas of all bases and faces of a three-                        i. sphere
                                              ?__.
           dimensional figure is called its ___
                                                                                          j. surface area
        8. The number of cubic units needed to fill a three-dimensional
           figure is called its ___
                                  ?__.                                                k. vanishing point
        9. A(n) ___ drawing shows three sides of a three-dimensional
                   ?__                                                                    l. volume
           figure.
                  ?__
       10. A(n) ___ drawing shows the individual sides of a three-
           dimensional figure.

LESSON 10-1              Visualize and Represent Solids, p. 422
                                                                                                 E          F
             A polyhedron is a closed, three-dimensional figure made up
             of polygonal surfaces called faces. A segment that is the
             intersection of two faces is an edge. The point at which          B
                                                                                           C
                                                                                                        D
             three or more edges intersect is a vertex.

       Use the figure at the right.                                                  A

       11. Identify the figure.                           12. Identify its base(s).
       13. Identify its lateral face(s).                  14. State the number of vertices.
       15. State the number of edges.                     16. Identify all edges that are skew to CF.

LESSON 10-2              Nets and Surface Area, p. 426
             A net is a two-dimensional pattern that can be folded to form a three-
             dimensional figure. A net can be used to find surface area.

       Draw a net for each three-dimensional figure. Then find the surface area.
       17.                                 18.       2.4 mm              19.                     6 cm
                                5m
                                                     4.6 mm
             12 m        18 m

                                                                                               4 cm
                                                                                   4 cm
464   Chapter 10 Three-Dimensional Geometry
LESSON 10-3             Surface Area of Three-Dimensional Figures, p. 432
           Surface areas can be found using these formulas.
           rectangular prism:              cylinder:                    cone:
           SA 2(lw lh wh)                  SA 2 r2       2 rh           SA    rs           r2

     Find the surface area of each figure. Round to the nearest hundredth.
     20.                  2.1 m            21.         8m               22.        5 ft


                       6.5 m                              20.2 m                          11 ft

           3.8 m


     23. A soup can has a height of 10 cm and a diameter of 6.5 cm. Find the amount
         of steel needed to make the can.
     24. A pet carrier is in the shape of a rectangular prism. It is 2.5 ft long, 1 ft high
         and 1.25 ft wide. What is the surface area of the carrier?

LESSON 10-4             Perspective Drawings, p. 436
           A perspective drawing is a way of drawing objects on a flat surface so that
           they look the same as they appear in real life.
     25. Sketch a rectangular tissue box in one-point perspective.
     26. Sketch a rectangular tissue box in two-point perspective.
     27. Locate the vanishing point in the perspective drawing.

LESSON 10-5             Isometric Drawings, p. 442
           An isometric drawing shows an object from a corner view so that
           three sides of the object can be seen in a single drawing.

     Give the number of cubes used to make each figure and the number of cube
     faces exposed. Be sure to count the cubes that are hidden from view.
     28.                                   29.                          30.




LESSON 10-6             Orthogonal Drawings, p. 446
           An orthogonal drawing gives top, front and side views of a three-dimensional
           figure as seen from a “straight on” viewpoint.

     Make an orthogonal drawing labeling the front, top and right-side views.
     31.                                   32.                          33.




                                                                                   Chapter 10 Review   465
       For each foundation drawing, sketch the front and right orthogonal views.
       34.                                    35.                            36.
              2    1   1                                 1                                             3

              1    2   3                            2    1                                             2

                                                    2    4                          3     4     1      1



LESSON 10-7                Volume of Prisms and Pyramids, p. 452
             Volume can be found using these formulas.
                                                               1
                  prism: V      Bh             pyramid: V      3
                                                                 Bh
       37. Find the volume of a square pyramid if the length of a side of its base is
           15 cm and its height is 23 cm.
       38. A rectangular prism has a base that measures 5 ft by 3 ft. Its volume is
           300 ft3. What is its height?
       39. A rectangular cake pan is 2 in.-by-13 in.-by-9 in. A round cake pan has
           a diameter of 8 in. and a height of 2 in. Which will hold more cake batter,
           the rectangular pan or two round pans?

LESSON 10-8                Volume of Cylinders, Cones and Spheres, p. 456
             Volume can be found using these formulas.
                                                             1 2                              4 3
                  cylinder: V        r2h       cone: V       3
                                                               rh            sphere: V        3
                                                                                                r

       Find the volume of each figure. Round to the nearest tenth.
       40. cylinder: r       6.1 mm, h     3.8 mm              41. cone: r     4.2 cm, h        11 cm
       42. sphere: r       6 mm                                43. cone: r     5 in., h       15 in.

LESSON 10-9                Problem Solving Skills: Length, Area and Volume,
                           p. 462
             The strategy, use an equation or formula, is appropriate when you have data
             that can be substituted into a formula or equation.
       44. A rectangular room measures 12 yd by 22 ft. How many square yards of
           carpet will it take to carpet the entire room?
       45. A rectangular room is 12 ft-by-21 ft. The walls are 8 ft tall. Paint is sold in
           1-gal containers. If a gallon of paint covers 450 ft2, how many gallons of
           paint should be purchased to paint the walls of the room?


CHAPTER INVESTIGATION
       EXTENSION Write a report about how the study of mathematics influences
       history. Your report could answer some of the following questions. How did
       cultural and historical events affect mathematics? How does geography affect
       the spread of information and new discoveries? How did mathematics help
       trigger major cultural events like the Industrial Revolution and the Information
       Age? Have all mathematicians been formally educated? How did computers
       change the way mathematicians work?

466   Chapter 10 Three-Dimensional Geometry
Chapter 10 Assessment
Use the figure to name the following.                                          E

 1. a pair of intersecting edges
                                                               B                   F
                                                                           D
 2. a pair of parallel edges
 3. a pair of parallel faces                              A           C
 4. a pair of edges that are skew
 5. the bases


Draw a net for each figure. Then identify the figure and find its surface area.
 6.                                         7.                                         8.
                                                 17 m
                              21 ft                                                                 16 cm

      3 ft
                                                  8m          8m
             10 ft                                                                                 9 cm


 9. Find the surface area of a cone with a radius of 3.1 cm and a slant height of
    12.4 cm.

10. Find the surface area of the figure.                           11. Locate the vanishing point of the figure.



                     2.5 cm




12. Create an isometric drawing of a figure composed of 5 cubes.

13. Make an orthogonal drawing of the figure you drew in Exercise 12. Show the
    front, top and right-side views.


Find the volume of each figure. Round to the nearest tenth if necessary.
14.                                   15.                                          16.
                       9m                                          5 in.
                                                                                            16 m
                                            13 in.        18 in.
                                                                                                          10 m

      4.5 m            6.2 m
                                                                                                          4m



17. To find the number of square feet of wrapping paper needed to cover a box
    shaped like a rectangular prism, which formula should you use?
       a. P          2l       2s            b. V        lwh                            c. SA       2(lw        lh   wh)

18. Write a problem that can be solved by applying the formula for the volume of
    a sphere. Provide an answer for your problem.

  mathmatters2.com/chapter_assessment                                                        Chapter 10 Assessment        467
Standardized Test Practice
 Part 1 Multiple Choice                               6. What is the slope of the line that passes
                                                         through A( 3, 2) and B(5, 4)?
Record your answers on the answer sheet                  (Lesson 6-2)
provided by your teacher or on a sheet of paper.           A
                                                                    4
                                                                                           B
                                                                                                    3
                                                                    3                               4
 1. Desiree’s test scores are 83, 75, 86, and 82.               3                               4
                                                           C                               D
                                                                4                               3
    If her teacher uses the mean, what score
    does she need on the fifth test in order to       7. Which line is not parallel to 2x               3y   5?
    have an average of 85? (Lesson 1-2)                  (Lesson 8-1)
     A   85                    B    90                    A     2x 3y 1
     C   99                    D    105                   B   2x 3y 2
                                                          C   4x 6y 5
 2. Of the people who buy raffle tickets, 500 win
    nothing, 1 wins $25, and 1 wins $100,000. If          D   3x 2y 5
    you were promoting the raffle and wanted to
    give a misleading statistic about the average     8. If 2x y 3 and x             y     1, what is the value
    winning, which measure of central tendency           of y? (Lesson 8-4)
    would you use? (Lesson 1-7)                           A     2                          B        1
     A   mean                                             C   1                            D    2
     B   median
     C   mode                                         9. Factor x2       4x     4. (Lesson 9-8)
                                                                         2
     D   range                                             A    (x 2)                    B  (x 2)2
                                                           C    x(x 4)                  D   (x 2)(x               2)
 3. Which expression is not represented by
    2x 1? (Lesson 2-3)                               10. How many faces does a pentagonal pyramid
     A  twice a number minus one                         have? (Lesson 10-1)
     B  twice a number less than one                      A  5                  B   6
     C  two times a number decreased by one               C  7                  D   8
     D  two times a number minus one
                                                     11. What is the volume of the
 4. Which graph represents the solution of               rectangular pyramid?
                                                         (Lesson 10-7)                                             11 ft
    3t 5     2? (Lesson 3-7)
      A
                                                          A   176 ft3
               3    2   1   0   1   2   3                 B   404 ft3                                             6 ft
      B                                                   C   528 ft3                               8 ft
                3   2   1   0   1   2   3
                                                          D   576 ft3
      C
               3    2   1   0   1   2   3

      D                                              Test-Taking Tip
                3   2   1   0   1   2   3
                                                     Question 11
                                                     Most standardized tests include any commonly used formulas
 5. What is the measure of each angle of a regular   at the front of the test booklet, but it will save you time to
    polygon that has 9 sides? (Lesson 5-7)           memorize many of these formulas. For example, you should
     A  126°                  B  140°                memorize that the volume of a pyramid is one-third the area
                                                     of the base times the height of the pyramid.
     C  150°                  D  180°

468       Chapter 10 Three-Dimensional Geometry
                                                                         Preparing for the Standardized Tests
                                                                         For test-taking strategies and more
                                                                         practice, see pages 627–644.


Part 2 Short Response/Grid In                            21. The perimeter of the rectangle is 16a 2b.
                                                             Write an expression for the length of the
Record your answers on the answer sheet                      rectangle. (Lessons 9-3 and 9-4)
provided by your teacher or on a sheet of paper.

                              2
12. What is the value of t        for t   5?                                                   5a – b
    (Lesson 2-8)

13. The area of a triangle can be determined by
                              1                          22. A bowling league has n teams. You can use
    using the formula A       2
                                bh. Solve this formula                       1      1
                                                             the expression n2        n to find the total
                                                                             2      2
    for h. (Lesson 3-2)                                      number of games that will be played if each
                                                             team plays each other team exactly once.
14. Solve 2(3t     6)   t    8. (Lesson 3-4)                 Factor this expression. (Lessons 9-7)

15. A bag contains 4 red marbles, 3 blue marbles,        23. What is the surface area of a cube with sides
    and 2 white marbles. One marble is chosen                of 7 in.? (Lesson 10-3)
    without replacement. Then another marble is
    chosen. What is the probability that the first
    marble is red and the second marble is blue?         24. A can of soup is 12 cm high and has a diameter
    (Lesson 4-5)                                             of 8 cm. A rectangular label is being designed
                                                             for this can of soup. If the label will cover the
                                                             surface of the can except for its top and
16. Find the value of x in circle Q. (Lesson 5-8)
                                                             bottom, what is the width and length of the
                              x°
                                                             label, to the nearest centimeter? (Lesson 10-9)

                             Q
                                                         Part 3 Extended Response
                                                         Record your answers on a sheet of paper.
17. The center of a circle is located at the origin      Show your work.
    of a coordinate plane. If A(5, 12) is on the
    circle, what is the radius of the circle?            25. Describe the three-dimensional figure.
    (Lesson 6-1)                                             Include the number of cubes needed to
                                                             make the figure and the number of cube
18. The distance a vehicle travels at a given speed          faces exposed. Draw an orthogonal drawing
    is a direct variation of the time it travels. If a       labeling the front, top, and right-side views.
    vehicle travels 30 mi in 45 min, how far can it          Then, draw a foundation drawing.
    travel in 2 h? (Lesson 6-8)                              (Lessons 10-5 and 10-6)

19. Triangle RST with vertices R(5, 4), S(3, 1),
    and T(0, 2) is translated so that R' is at (3, 1).
    What are the coordinates of S' ?
    (Lesson 7-1)

20. Solve the system of equations.
                                                         26. Draw a cylinder and a cone that have the
    (Lesson 8-3)
                                                             same volume. Explain your response and
              y 3x                                           calculate the volume and surface area of each
              x 2y       21                                  figure. (Lesson 10-8)

            mathmatters2.com/standardized_test                     Chapter 10 Standardized Test Practice       469