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					10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones



    Warm Up – No Warm up. Better know
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    Take out your homework.




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones



                       Objectives
   Learn and apply the formula for the
   volume of a prism.
   Learn and apply the formula for the
   volume of a cylinder.




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones



                      Vocabulary
   volume




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones


  The volume of a three-dimensional figure is the
  number of nonoverlapping unit cubes of a given size
  that will exactly fill the interior.
  Cavalieri’s principle says that if two three-
  dimensional figures have the same height and have
  the same cross-sectional area at every level, they
  have the same volume.
                                       A right prism and
                                       an oblique prism
                                       with the same base
                                       and height have the
                                       same volume.


Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                    Example 1 Continued

  Find the volume of the right
  regular hexagonal prism. Round
  to the nearest tenth, if
  necessary.




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones


 Cavalieri’s principle
 also relates to cylinders.
 The two stacks have
 the same number of
 CDs, so they have
 the same volume.




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                    Example 3 Continued

  Find the volume of a cylinder with base area 
  and a height equal to twice the radius. Give
  your answers in terms of  and rounded to the
  nearest tenth.
  Step 3 Use the radius and height to find the volume.




     = 2662 cm3  8362.9 cm3



Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

Example 4: Exploring Effects of Changing Dimensions


  The radius and height of the
  cylinder are multiplied by .
  Describe the effect on the
  volume.


                                  radius and height
    original dimensions:
                                  multiplied by :




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                      Example 4 Continued


  The radius and height of the
  cylinder are multiplied by .
  Describe the effect on the
  volume.


  Notice that                     . If the radius and
  height are multiplied by    , the volume is multiplied
  by       , or   .



Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                          Example 5

  Find the volume of the composite
  figure. Round to the nearest tenth.


  Find the side length s of the base:
  The volume of the                   The volume of
  square prism is:                    the cylinder is:


  The volume of the composite is the cylinder minus
  the rectangular prism.
        Vcylinder — Vsquare prism = 45 — 90  51.4 cm3
Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones



                       Objectives
   Learn and apply the formula for the
   volume of a pyramid.
   Learn and apply the formula for the
   volume of a cone.




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones


   The square pyramids are congruent, so they have the
   same volume. The volume of each pyramid is one
   third the volume of the cube.




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

         Example 6: Finding Volumes of Pyramids

    Find the volume a rectangular pyramid with
    length 11 m, width 18 m, and height 23 m.




Holt Geometry
10-6 & 7 Volume    of Prisms, Cylinders, Pyramids and Cones

                     Example 7 Continued

   Find the volume of the regular
   hexagonal pyramid with height
   equal to the apothem of the base




      = 1296 ft3
Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                    Example 9 Continued

   Find the volume of a cone with base
   circumference 25 in. and a height 2 in. more
   than twice the radius.




            = 1406.25 in3 ≈ 4417.9 in3


Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                   Example 10 Continued

      Find the volume of this cone.




          2560 cm3  8042.5 cm3
Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

  Example 11: Exploring Effects of Changing Dimensions

  The diameter and height of the
  cone are divided by 3. Describe
  the effect on the volume.

 original dimensions:        radius and height divided by 3:




  Notice that                . If the radius and height
  are divided by 3, the volume is divided by 33, or 27.
Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

  Example 12: Finding Volumes of Composite Three-
                Dimensional Figures

   Find the volume of the composite
   figure. Round to the nearest tenth.
   The volume of the cylinder is
   Vcylinder = r2h = (21)2(35)=15,435 cm3.

   The volume of the lower cone is



   The volume of the figure is the sum of the volumes.
  V = 5145 + 15,435 + 5,880 = 26,460  83,126.5 cm3
Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                         Lesson Quiz:

    1. Find the volume of a cylinder with base area
    196 cm2 and a height equal to the diameter
                            V  17,241.1 cm3

    2. The edge length of the cube is tripled.
       Describe the effect on the volume.
       The volume is multiplied by 27.

    3. Find the volume of the composite
       figure. Round to the nearest tenth.
       9160.9 in3

Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                         Lesson Quiz:

    4. A cone has radius 2 in. and height 7 in. If the
       radius and height are multiplied by , describe
       the effect on the volume.
       The volume is multiplied by    .
    5. Find the volume of the composite figure. Give
       your answer in terms of .
       10,800 yd3




Holt Geometry
10-6 & 7 Volume   of Prisms, Cylinders, Pyramids and Cones

                       Homework
    • Pg 702 #14, 18, 20-23, 30, 35, 41-44
    • Pg 710 #14, 17, 21-27, 33-36, 39, 46-50

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Holt Geometry

				
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