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Areas of Polygons and Circles
Chapter
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2. Circumference and Area of
a Circle
8.4
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3. Circumference and Area of a Circle
Theorem The circumference of a circle is given by the formula C =  d or C = 2 r

4. Value of 
When a calculator is used to determine  with greater accuracy, we see an approximation such as  =

5. Example 1
In O in Figure 8.41, OA = 7 cm. Using   a) find the approximate circumference C of O. b) find the approximate length of the minor arc . Solution: a) C = 2 r = = 44 cm
Figure 8.41

6. Example 1 – Solution
cont’d
b) Because the degree of measure of is 90, the arc length is of the circumference, 44 cm. Thus, length of =

7. LENGTH OF AN ARC

8. Length of an Arc
Informally, the length of an arc is the distance between the endpoints of the arc as though it were measured along a straight line. Two further considerations regarding the measurement of arc length follow. 1. The ratio of the degree measure m of the arc to 360 (the degree measure of the entire circle) is the same as the ratio of the length ℓ of the arc to the circumference; that is,

9. Length of an Arc
2. Just as m denotes the degree measure of an arc, ℓ denotes the length of the arc. Whereas m is measured in degrees, ℓ is measured in linear units such as inches, feet, or centimeters.

10. Length of an Arc
Theorem In a circle whose circumference is C, the length ℓ of an arc whose degree measure is m is given by Note: For arc AB,

11. Example 4
Find the approximate length of major arc ABC in a circle of radius 7 in. if = 45. See Figure Use   .
Figure 8.43

12. Example 4 – Solution
According to Theorem 8.4.2, or which can be simplified to

13. AREA OF A CIRCLE

14. Area of a Circle
Theorem The area A of a circle whose radius has length r is given by A =  r 2.

15. Example 6
Find the approximate area of a circle whose radius has a length of 10 in. Use   3.14.
Solution:
A =  r2
becomes A = 3.14(10)2.
Then
A = 3.14(100)
= 314 in2