Lesson 8

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					Lesson 8
 Perimeter and Area




                      1
               Perimeter
• The perimeter of a closed figure
  is the distance around the
  outside of the figure.
• In the case of a polygon, the
  perimeter is found by adding the
  lengths of all of its sides. No
  special formulas are needed.
• Units for perimeter include
  inches, centimeters, miles, etc.
                                     2
                   Example
• In the figure, ABCD is a
  rectangle, AB  6 and          D   C
  BC  8. What is the
  perimeter of this rectangle?
• The opposite sides of a
  rectangle are congruent.
  So, CD  6 and AD  8.
• So, the perimeter is
                                 A   B
    6  8  6  8  28.              3
            Circumference
• The distance around the outside of a circle
  is traditionally called the circumference of
  the circle, not the perimeter.
• The circumference of a circle with
  diameter d is given by the formula
                C   d
• Here,  (pi) is a mathematical constant
  equal to approximately 3.14159265.
                                             4
                 Example
• The radius of a circle is 5 inches.
• What is the circumference of this circle?
  Round to the nearest hundredth of an inch.
• First, note that
         d  2r  2  5  10 inches.
• So,
         C    d  10  31.42 inches.
• Note: scientific calculators have a 
  button.                                  5
                      Example
• In the figure, a rectangle is surmounted
  by a semicircle.
• Given the measurements as marked,
  find the perimeter of the figure.
                                                5
• Note that the diameter of the circle is
  10. So, the top and bottom sides of the       10
  rectangle are also 10.
• The left side of the rectangle is 15.    15        15
• The circumference of the semicircle is
           0.5   10  5.
• So, the perimeter of the figure is            10

    15 10 15  5  40 5.                         6
                    Area
• The area of a closed figure (like a circle or
  a polygon) measures the amount of
  “space” the figure takes up.
• For example, to find out how much carpet
  to order for a room, you would need to
  know the area of the room’s floor.
• Units used for area include square
  centimeters  (cm2 ), square miles, square
  yards, and acres.
                                                  7
          Area of a Rectangle
• The area of a rectangle is found by
  multiplying its base times its height
  (or length times width).
• If the base is b and the height is h
  as in the figure, then the area
  formula is                                      h

              A  b h
• Note that the product of two length
                                          b
  units gives area units (like: inches
  times inches equals square inches).
                                              8
                       Example
• The figure shown is a square whose
  diagonal measures 10.
• What is the area of the square?
• Using our knowledge of 45-45-90
  triangles, note that each side of the
  square must be                                      5 2
         10 10 2 10 2                            10
                            5 2
          2     2 2       2
• Now, since a square is a rectangle, we        5 2
  find its area by multiplying base times
  height:
      A  5 2  5 2  25 2  2  25  2  50.         9
          Altitudes of Triangles
• The formula for the area of a triangle involves the length
  of an altitude of the triangle. So, first we discuss what an
  altitude is.
• An altitude is a line segment that runs from one vertex of
  the triangle to the opposite side or extension of the
  opposite side, and it is perpendicular to this opposite side
  (or extension).
• Some altitudes are drawn below and marked h:



    h             h                   h

                                                           10
          Area of a Triangle
• The formula for the area of a triangle is
                       1
                   A  bh
                       2
  where b is the length of the base (one of
  the sides of the triangle) and h is the
  height (the length of the altitude drawn to
  the base).

                                                11
                       Example
• The triangle in the figure is a right triangle
  with right angle at A, and sides as
  marked.
• Find the area of this triangle.
• We will take AB as the base. Then the           A   20         C
  height would be AC, which we can find
  with the Pythagorean Theorem:                  15
                                                       25
  AC  25 15  625  225  400  20.
            2    2


• So, the area is:                               B
             1
          A  15  20  15 10  150.
             2
                                                            12
        Area of a Parallelogram
• To find the area of a parallelogram
  multiply its base times its height.
• The base is any side of the
  parallelogram like the one marked b in
  the figure.
• The height is the length of an altitude
  drawn to the base like the one marked
  h in the figure.


                  h

                                            13
                      b
           Area of a Trapezoid
• To find the area of a trapezoid multiply the height by the
  mean of the two bases.
• If the height is h and the bases are b and B as in the
  figure, then the area formula is:
                           bB
                     A  h   
                            2 

                             b

                        h

                                                           14
                             B
            Area of a Circle
• The area of a circle with radius r is found
  by multiplying pi by the radius squared.
• The formula is
                  A  r   2




                                                15
               Heron’s Formula
• Heron’s Formula is used to find
  the area of a triangle when
  altitudes are unknown, but all
  three sides are known.
• If the lengths of the sides of the
  triangle are a, b, and c, then the   a       b
  area is given by the formula

     A  s(s  a)(s  b)(s  c)
                                           c
  where s is the semiperimeter:
            1
         s  (a  b  c)
            2                                      16
             Adding Areas
• If you have to find the area of a complex
  shape, try dissecting the shape into non-
  overlapping simple shapes that you can
  find the area of. Then add the areas of the
  simple shapes.
• For example, note how the shape below is
  dissected into two rectangles.



                                            17
             Subtracting Areas
• Sometimes the area of a complex figure, especially one
  with “holes” in it, can be found by subtracting the areas of
  simpler figures.
• For example, to find the shaded area below, we would
  subtract the area of the circle from the area of the
  rectangle.




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