Copyright © Cengage Learning. All rights reserved. 8 Areas of Polygons and Circles Chapter Copyright © Cengage Learning. All rights reserved.

Circumference and Area of a Circle 8.4 Copyright © Cengage Learning. All rights reserved.

Circumference and Area of a Circle Theorem 8.4.1 The circumference of a circle is given by the formula C =  d or C = 2 r

Value of  When a calculator is used to determine  with greater accuracy, we see an approximation such as  = 3.141592654.

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

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 =

LENGTH OF AN ARC

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,

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.

Length of an Arc Theorem 8.4.2 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,

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

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

AREA OF A CIRCLE

Area of a Circle Theorem 8.4.3 The area A of a circle whose radius has length r is given by A =  r 2.

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