1. home and garden
2. home furnishings
3. sofas and chairs

4.4
Area and Circumference
4.4
OBJECTIVES
1.
2.
3.
4.
5.
Use p to find the circumference of a circle
Use p to find the area of a circle
Find the area of a parallelogram
Find the area of a triangle
Convert square units
In Section 4.2, we again looked at the perimeter of a straight-edged figure. The distance
around the outside of a circle is closely related to this concept of perimeter. We call the
perimeter of a circle the circumference.
Definitions: Circumference of a Circle
The circumference of a circle is the distance around that circle.
Radius
O
d
Diameter
Circumference
Figure 1
Let’s begin by defining some terms. In the circle of Figure 1, d represents the diameter.
This is the distance across the circle through its center (labeled with the letter O, for origin). The radius r is the distance from the center to a point on the circle. The diameter is
always twice the radius.
It was discovered long ago that the ratio of the circumference of a circle to its diameter
always stays the same. The ratio has a special name. It is named by the Greek letter p (pi).
22
Pi is approximately , or 3.14 rounded to two decimal places. We can write the following
7
formula.
© 2001 McGraw-Hill Companies
NOTE The formula comes from
the ratio
C
p
d
Rules and Properties: Formula for the Circumference of
a Circle
C pd
4.5 ft
(1)
Example 1
Finding the Circumference of a Circle
Figure 2
A circle has a diameter of 4.5 ft, as shown in Figure 2. Find its circumference, using 3.14
for p. If your calculator has a p key, use that key instead of a decimal approximation
for p.
341
342
CHAPTER 4
DECIMALS
NOTE Because 3.14 is an
approximation for pi, we can
only say that the circumference
is approximately 14.1 ft. The
symbol means approximately.
By Formula (1),
C pd
3.14 4.5 ft
14.1 ft (rounded to one decimal place)
CHECK YOURSELF 1
approximate p, you needn’t
worry about running out of
decimal places. The value for pi
has been calculated to over
100,000,000 decimal places on a
computer (the printout was
some 20,000 pages long).
NOTE Because d 2r (the
diameter is twice the radius)
and C pd, we have C p(2r),
or C 2pr.
1
A circle has a diameter of 3 inches (in.). Find its circumference.
2
Note: In finding the circumference of a circle, you can use whichever approximation for pi
you choose. If you are using a calculator and want more accuracy, use the p key.
There is another useful formula for the circumference of a circle.
Rules and Properties:
Formula for the Circumference of a Circle
C 2pr
(2)
Example 2
Finding the Circumference of a Circle
A circle has a radius of 8 in., as shown in Figure 3. Find its circumference using 3.14 for p.
8 in.
Figure 3
From Formula (2),
C 2pr
2 3.14 8 in.
50.2 in. (rounded to one decimal place)
CHECK YOURSELF 2
Find the circumference of a circle with a radius of 2.5 in.
© 2001 McGraw-Hill Companies
NOTE If you want to
AREA AND CIRCUMFERENCE
SECTION 4.4
343
Sometimes we will want to combine the ideas of perimeter and circumference to solve
a problem.
Example 3
Finding Perimeter
We wish to build a wrought-iron frame gate according to the diagram in Figure 4. How
many feet (ft) of material will be needed?
NOTE The distance around the
semicircle is
1
pd.
2
5 ft
4 ft
Figure 4
The problem can be broken into two parts. The upper part of the frame is a semicircle (half
a circle). The remaining part of the frame is just three sides of a rectangle.
NOTE Using a calculator with a
Circumference (upper part) p key,
1 2 p 5
1
3.14 5 ft 7.9 ft
2
Perimeter (lower part) 4 5 4 13 ft
Adding, we have
7.9 13 20.9 ft
We will need approximately 20.9 ft of material.
CHECK YOURSELF 3
Find the perimeter of the following figure.
© 2001 McGraw-Hill Companies
6 yd
8 yd
The number pi (p), which we used to find circumference, is also used in finding the area
of a circle. If r is the radius of a circle, we have the following formula.
344
CHAPTER 4
DECIMALS
Rules and Properties:
A pr
NOTE This is read, “Area
Formula for the Area of a Circle
2
(3)
equals pi r squared.” You can
multiply the radius by itself and
then by pi.
Example 4
Find the Area of a Circle
A circle has a radius of 7 inches (in.) (see Figure 5). What is its area?
7 in.
Figure 5
Use Formula (3), using 3.14 for p and r 7 in.
A 3.14 (7 in.)2
Again the area is an approximation
because we use 3.14, an approximation
for p.
153.86 in.2
CHECK YOURSELF 4
Find the area of a circle whose diameter is 4.8 centimeters (cm). Remember that the
formula refers to the radius. Use 3.14 for p, and round your result to the nearest
tenth of a square centimeter.
Two other figures that are frequently encountered are parallalograms and triangles.
B
C
h
b
D
Figure 6
In Figure 6 ABCD is called a parallelogram. Its opposite sides are parallel and equal.
Let’s draw a line from D that forms a right angle with side BC. This cuts off one corner of
the parallelogram. Now imagine that we move that corner over to the left side of the figure,
as shown. This gives us a rectangle instead of a parallelogram. Because we haven’t changed
the area of the figure by moving the corner, the parallelogram has the same area as the
rectangle, the product of the base and the height.
© 2001 McGraw-Hill Companies
A
AREA AND CIRCUMFERENCE
SECTION 4.4
345
Rules and Properties: Formula for the Area of
a Parallelogram
Abh
(4)
Example 5
Finding the Area of a Parallelogram
A parallelogram has the dimensions shown in Figure 7. What is its area?
1.8 in.
3.2 in.
Figure 7
Use Formula (4), with b 3.2 in. and h 1.8 in.
Abh
3.2 in. 1.8 in. 5.76 in.2
CHECK YOURSELF 5
1
1
If the base of a parallelogram is 3 in. and its height is 1 in., what is its area?
2
2
Another common geometric figure is the triangle. It has three sides. An example is
triangle ABC in Figure 8.
B
D
b is the base of the triangle.
h is the height, or the
altitude, of the triangle.
© 2001 McGraw-Hill Companies
h
A
b
C
Figure 8
Once we have a formula for the area of a parallelogram, it is not hard to find the area of a
triangle. If we draw the dotted lines from B to D and from C to D parallel to the sides of
the triangle, we form a parallelogram. The area of the triangle is then one-half the area
of the parallelogram [which is b h by Formula (4)].
346
CHAPTER 4
DECIMALS
Rules and Properties:
A
Formula for the Area of a Triangle
1
b h
2
(5)
Example 6
Finding the Area of a Triangle
A triangle has an altitude of 2.3 in., and its base is 3.4 in. (see Figure 9). What is its area?
2.3 in.
3.4 in.
Figure 9
Use Formula (5), with b 3.4 in. and h 2.3 in.
A
1
b h
2
1
3.4 in. 2.3 in. 3.91 in.2
2
CHECK YOURSELF 6
A triangle has a base of 10 feet (ft) and an altitude of 6 ft. Find its area.
Sometimes we will want to convert from one square unit to another. For instance, look
at 1 yd2 in Figure 10.
1 yd = 3 ft
1 yd 2 = 9 ft 2
Figure 10
The table below gives some useful relationships.
Square Units and Equivalents
NOTE Originally the acre was
the area that could be plowed
by a team of oxen in a day!
1 square foot (ft2) 144 square inches (in.2)
1 square yard (yd2) 9 ft2
1 acre
4840 yd2 43,560 ft2
© 2001 McGraw-Hill Companies
1 yd = 3 ft
AREA AND CIRCUMFERENCE
SECTION 4.4
347
Example 7
Converting Between Feet and Yards in Finding Area
A room has the dimensions 12 ft by 15 ft. How many square yards of linoleum will be
needed to cover the floor?
1 12
NOTE We first find the area in
A 12 ft 15 ft 180 ft2
square feet, then convert to
square yards.
20
180 ft2 1 yd2
9 ft2
20 yd2
CHECK YOURSELF 7
A hallway is 27 ft long and 4 ft wide. How many square yards of carpeting will be
needed to carpet the hallway?
Example 8 illustrates the use of a common unit of area, the acre.
Example 8
Converting Between Yards and Acres in Finding Area
A rectangular field is 220 yd long and 110 yd wide. Find its area in acres.
A 220 yd 110 yd 24,200 yd2
© 2001 McGraw-Hill Companies
5
24,200 yd2
5 acres
1 acre
4840 yd2
1
CHECK YOURSELF 8
A proposed site for an elementary school is 220 yd long and 198 yd wide. Find its
area in acres.
CHAPTER 4
DECIMALS
CHECK YOURSELF ANSWERS
1. C 11 in.
2. C 15.7 in.
3. P 31.4 yd
1
1
1
5. A 3 in. 1 in.
6. A 10 ft 6 ft
2
2
2
7
3
in. in.
= 30 ft2
2
2
1
5 in.2
4
4. 18.1 cm2
7. 12 yd2
8. 9 acres
© 2001 McGraw-Hill Companies
348
Name
4.4 Exercises
Section
Date
Find the circumference of each figure. Use 3.14 for p, and round your answer to one
decimal place.
ANSWERS
1.
2.
1.
5 ft
9 ft
2.
3.
4.
3.
5.
4.
8.5 in.
6.
3.75 ft
7.
8.
9.
In exercises 5 and 6, use
5.
22
for p, and find the circumference of each figure.
7
10.
6.
1
1
17 2 in.
3 2 ft
Find the perimeter of each figure. (The curves are semicircles.) Round answers to one
decimal place.
7.
8.
9 ft
© 2001 McGraw-Hill Companies
7 ft
3 in.
1 in.
9.
10.
4 ft
7 ft
10 in.
349
ANSWERS
11.
Find the area of each figure. Use 3.14 for p, and round your answer to one decimal place.
12.
11.
12.
13.
12 ft
7 in.
14.
15.
16.
13.
17.
14.
7 yd
18.
8 ft
19.
20.
In exercises 15 and 16, use
22
for p, and find the area of each figure.
7
15.
16.
1
1
3 2 yd
1 2 in.
Find the area of each figure.
17.
18.
4 ft
8 in.
4 in.
19.
20.
3 yd
4 yd
350
5 in.
7 in.
© 2001 McGraw-Hill Companies
7 ft
ANSWERS
21.
21.
22.
22.
5 ft
6 ft
23.
8 ft
11 ft
24.
25.
23.
24.
26.
27.
13 yd
12 in.
28.
13 yd
29.
9 in.
25.
2 ft
26.
6 yd
4 ft
5 ft
7 yd
2 yd
Solve the following applications.
27. Jogging. A path runs around a circular lake with a diameter of 1000 yards (yd).
Robert jogs around the lake three times for his morning run. How far has he run?
28. Binding. A circular rug is 6 feet (ft) in diameter. Binding for the edge costs $1.50
© 2001 McGraw-Hill Companies
per yard. What will it cost to bind around the rug?
29. Lawn care. A circular piece of lawn has a radius of 28 ft. You have a bag of fertilizer
that will cover 2500 ft2 of lawn. Do you have enough?
351
ANSWERS
30.
30. Cost. A circular coffee table has a diameter of 5 ft. What will it cost to have the top
refinished if the company charges $3 per square foot for the refinishing?
31.
32.
33.
34.
35.
36.
37.
38.
31. Cost. A circular terrace has a radius of 6 ft. If it costs $1.50 per square foot to pave
the terrace with brick, what will the total cost be?
32. Area. A house addition is in the shape of a semicircle (a half circle) with a radius of
9 ft. What is its area?
33. Amount of material. A Tetra-Kite uses 12 triangular pieces of plastic for its surface.
Each triangle has a base of 12 inches (in.) and a height of 12 in. How much material
is needed for the kite?
34. Acreage. You buy a square lot that is 110 yd on each side. What is its size in acres?
material will you need for four posters?
36. Cost. Andy is carpeting a recreation room 18 feet (ft) long and 12 ft wide. If the
carpeting costs $15/yd2, what will be the total cost of the carpet?
37. Acreage. A shopping center is rectangular, with dimensions of 550 by 440 yd. What
is its size in acres?
38. Cost. An A-frame cabin has a triangular front with a base of 30 ft and a height of
20 ft. If the front is to be glass that costs $3 per square foot, what will the glass cost?
352
© 2001 McGraw-Hill Companies
35. Area. You are making rectangular posters 12 by 15 in. How many square feet of
ANSWERS
Find the area of the shaded part in each figure. Round your answers to one decimal place.
39.
40.
Semicircle
39.
40.
41.
3 ft
2 ft
5 ft
43.
6 ft
41.
42.
42.
10 in.
20 ft
44.
10 in.
20 ft
45.
43. Papa Doc’s delivers pizza. The 8-inch (in.)-diameter pizza is $8.99, and the price of
a 16-in.-diameter pizza is $17.98. Write a plan to determine which is the better
buy.
46.
44. The distance from Philadelphia to Sea Isle City is 100 miles (mi). A car was driven
this distance using tires with a radius of 14 in. How many revolutions of each tire
occurred on the trip?
45. Find the area and the circumference (or perimeter) of each of the following:
46. An indoor track layout is shown below.
7
m
© 2001 McGraw-Hill Companies
(a) a penny (b) a nickel (c) a dime (d) a quarter (e) a half-dollar (f) a silver
dollar (g) a Susan B. Anthony dollar (h) a dollar bill (i) one face of the pyramid
on the back of a $1 bill.
20 m
How much would it cost to lay down hardwood floor if the hardwood floor costs
$10.50 per square meter?
353
ANSWERS
47. What is the effect on the area of a triangle if the base is doubled and the altitude is
cut in half? Create some examples to demonstrate your ideas.
47.
48.
48. How would you determine the cross-sectional area of a Douglas fir tree (at, say, 3 ft
above the ground), without cutting it down? Use your method to solve the following
problem:
49.
50.
If the circumference of a Douglas fir is 6 ft 3 in., measured at a height of 3 ft above
the ground, compute the cross-sectional area of the tree at that height.
49. What happens to the circumference of a circle if you double the radius? If you double
the diameter? If you triple the radius? Create some examples to demonstrate your
answers.
50. What happens to the area of a circle if you double the radius? If you double the
diameter? If you triple the radius? Create some examples to demonstrate your
answers.
Answers
11.
21.
29.
37.
47.
354
3. 26.7 in.
7. 37.1 ft
9. 34.6 ft
5 2
153.9 in.
13. 38.5 yd
15. 9 yd
17. 28 ft2
19. 12 yd2
8
20 ft2
23. 54 in.2
25. 24 ft2
27. 9420 yd
2
Yes; area 2461.8 ft
31. $169.56
33. 864 in.2
35. 5 ft2
2
2
50 acres
39. 50.2 ft
41. 86 ft
43.
45.
49. Doubled; doubled; tripled
2
5. 55 in.
2
© 2001 McGraw-Hill Companies
1. 56.5 ft