Chapter 8 Areas of Polygons and Circles Copyright

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

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

8. 4 Circumference and Area of a Circle Copyright © Cengage Learning. All rights

8. 4 Circumference and Area of a Circle Copyright © Cengage Learning. All rights reserved.

Circumference and Area of a Circle Theorem 8. 4. 1 The circumference of a

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 3

Value of When a calculator is used to determine with greater accuracy, we see

Value of When a calculator is used to determine with greater accuracy, we see an approximation such as = 3. 141592654. 4

Example 1 In O in Figure 8. 41, OA = 7 cm. Using a)

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 5

Example 1 – Solution cont’d b) Because the degree of measure of is 90

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 = 6

LENGTH OF AN ARC 7

LENGTH OF AN ARC 7

Length of an Arc Informally, the length of an arc is the distance between

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, 8

Length of an Arc 2. Just as m denotes the degree measure of an

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. 9

Length of an Arc Theorem 8. 4. 2 In a circle whose circumference is

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, 10

Example 4 Find the approximate length of major arc ABC in a circle of

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 11

Example 4 – Solution According to Theorem 8. 4. 2, or which can be

Example 4 – Solution According to Theorem 8. 4. 2, or which can be simplified to 12

AREA OF A CIRCLE 13

AREA OF A CIRCLE 13

Area of a Circle Theorem 8. 4. 3 The area A 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. 14

Example 6 Find the approximate area of a circle whose radius has a length

Example 6 Find the approximate area of a circle whose radius has a length of 10 in. Use 3. 14. Solution: A = r 2 becomes A = 3. 14(10)2. Then A = 3. 14(100) = 314 in 2 15