Lesson 8

Document Sample

```					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
• 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.

18

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 10 posted: 5/8/2010 language: English pages: 18