8.7 Circumference and Area of Circles
Page 1 of 8
8.7 Circumference and Area
of Circles
Goal
Find the circumference and area of circles.
Key Words
? circle ? center ? radius ? diameter ? circumference ? central angle ? sector
A circle is the set of all points in a plane that are the same distance from a given point, called the center of the circle. A circle with center P is called "circle P," or P.
The distance from the center to a point on the circle is the radius . The plural of radius is radii.
The distance across the circle, through the center, is the diameter . The diameter d is twice the radius r. So, d 2r.
radius diameter
center
The circumference of a circle is the distance around the circle.
circumference
For any circle, the ratio of the circumference to its diameter is denoted by the Greek letter , or pi. The number is 3.14159 . . . , which is an irrational number. This means that neither terminates nor repeats. So, an approximation of 3.14 is used for .
CIRCUMFERENCE OF A CIRCLE
Words Circumference (diameter)
2(radius)
r
Symbols C d or C 2r
EXAMPLE 1 Find the Circumference of a Circle
Find the circumference of the circle. 4 in.
Student Help
STUDY TIP When simplifying an expression involving , substitute 3.14 for . You can also use the key on your calculator, as in Examples 2 and 3.
Solution C 2r 2(4) 8 8(3.14) 25.12
Formula for the circumference Substitute 4 for r. Simplify.
Use 3.14 as an approximation for .
Multiply.
ANSWER The circumference is about 25 inches.
452 Chapter 8 Polygons and Area
Page 2 of 8
Find the Circumference of a Circle
Find the circumference of the circle. Round your answer to the nearest whole number.
1.
2.
3.
6 cm 9 ft
16 in.
AREA OF A CIRCLE
Words Area (radius)2
r
Symbols A r2
Student Help
KEYSTROKE HELP If your calculator has written above a key, use the following keystrokes to simplify 49:
49
EXAMPLE 2 Find the Area of a Circle
Find the area of the circle. 7 cm
Solution A r 2 (7)2 49 153.94
Formula for the area of a circle Substitute 7 for r. Simplify. Use a calculator.
ANSWER The area is about 154 square centimeters.
EXAMPLE 3 Use the Area of a Circle
Find the radius of a circle with an area of 380 square feet.
Solution
r A 380 ft2
A r 2 Formula for the area of a circle
380 r 2 Substitute 380 for A.
120.96 r 2
Divide each side by . Use a calculator.
11 r
Take the positive square root.
ANSWER The radius is about 11 feet.
8.7 Circumference and Area of Circles 453
Page 3 of 8
Find the Area of a Circle
Find the area of the circle. Round your answer to the nearest whole number.
4. 8 in.
5. 3 cm
6. 12 ft
Student Help
VOCABULARY TIP The term radius is also used to name a segment that connects the center of a circle to a point on the circle. Two such radii are used to determine a sector of a circle.
Visualize It!
A circle contains two straight angles. So, there are 180 180, or 360, in a circle.
180
360
or
180
Central Angles An angle whose vertex is the center of a circle is a central angle of the circle.
A region of a circle determined by two radii and a part of the circle is called a sector of the circle.
Because a sector is a portion of a circle, the following proportion can be used to find the area of a sector.
Area of sector
Measure of central angle
Area of entire circle Measur e of entir e circle
sector
central angle
EXAMPLE 4 Find the Area of a Sector
Find the area of the blue sector.
9 m 120
Solution
1 First find the area of the circle.
A r 2 (9)2 254.47
The area of the circle is about 254 square meters.
2 Then find the area of the sector. Let x equal the area of the sector.
Area of sector
Measure of central angle
Area of entire circle Measur e of entir e circle
25x4 132600
Substitute.
360x 254 p 120 Cross product property
360x 30,480
Simplify.
3366 00x 303,64 080
Divide each side by 360.
x 84.67
Simplify.
ANSWER The area of the sector is about 85 square meters.
454 Chapter 8 Polygons and Area
Page 4 of 8
Find the Area of a Sector
In Exercises 7 and 8, A represents the area of the entire circle and x represents the area of the blue sector. Complete the proportion used to find x. Do not solve the proportion.
7. A 22 m2
8. A 28 ft2
x? 18?0
180
x? 36?0
170
Find the area of the blue sector. Round your answer to the nearest whole number.
9.
2 ft 60
10. 5 cm
90
11.
6 in. 135
8.7 Exercises
Guided Practice
Vocabulary Check
1. Sketch a circle. Sketch and label a radius and a diameter of the circle.
2. Describe how to find the circumference of a circle given its radius.
Skill Check Copy and complete the table below.
Radius, r Diameter, d
r
3.
?
14 in.
4. 11 cm
?
d
5.
3.5 m
?
6.
?
1 ft
Write an equation for the area A or the circumference C by filling in
the missing number.
7. C 2 ?
8. A ? (3)2
9. A ( ? )2
8
3
14
8.7 Circumference and Area of Circles 455
Page 5 of 8
Practice and Applications
Extra Practice
See p. 690.
Finding Circumference Find the circumference of the circle. Round your answer to the nearest whole number.
10. 7 cm
11.
12.
10 m
2 in.
13. A circle with a radius of 13 yards
14. A circle with a diameter of 15 meters
Finding Area Find the area of the circle. Round your answer to the nearest whole number.
15. 4 ft
16. 1 cm
17. 6 in.
18. 9 m
19. 16 ft
20. 19 yd
21. Cranberries To harvest cranberries, the field is flooded so that the berries float. What area of cranberries can be gathered into a circular region with a radius of 5.5 meters?
22. Error Analysis A student was asked to find the area of the circle below. Describe any errors.
A r2 (3.14)(12)2 12 452.16
Homework Help
Example 1: Exs. 10?14
23. You be the Judge One of your
classmates states that if the radius of a
5
Example 2: Exs. 15?20 Example 3: Exs. 24?29
circle is doubled, then its area doubles.
10
Example 4: Exs. 31?35
Do you agree or disagree? Explain your
reasoning using the circles at the right.
456 Chapter 8 Polygons and Area
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