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Copyright © Cengage Learning. All rights reserved. 8 Locus and Concurrence Chapter Copyright © Cengage Learning. All rights reserved.

More Area Relationships in the Circle 8.5 Copyright © Cengage Learning. All rights reserved.

More Area Relationships in the circle Definition A sector of a circle is a region bounded by two radii of the circle and an arc intercepted by those radii. (See Figure 8.47.) Figure 8.47

AREA OF A SECTOR

Area of a Sector Postulate 23 The ratio of the degree measure m of the arc (or central angle) of a sector to 360° is the same as the ratio of the area of the sector to the area of the circle; that is, Theorem 8.5.1 In a circle of radius r, the area A of a sector whose arc has degree measure m is

Example 1 If m O = 100°, find the area of the 100° sector shown in Figure 8.49. Use your calculator and round the answer to the nearest hundredth of a square inch. Figure 8.49

Example 1 – Solution becomes  87.27 in2

Area of a Sector In applications with circles, we often seek exact answers for circumference and area; in such cases, we simply leave  in the result. For instance, in a circle of radius length 5 in., the exact circumference is 10 in. and the exact area is expressed as 25 in².

Area of a Sector Because a sector is bounded by two radii and an arc, the perimeter of a sector is the sum of the lengths of the two radii and the length of its arc. Corollary 8.5.2 The area of a semicircular region of radius length r is A =  r 2.

AREA OF A SEGMENT

Area of a Segment Definition A segment of a circle is a region bounded by a chord and its minor (or major) arc. In Figure 8.51, the segment is bounded by chord and its minor arc . Figure 8.51

Example 4 Find the exact area of the segment bounded by the chord and the arc whose measure is 90°. The radius has length 12 in., as shown in Figure 8.52. Figure 8.52

Example 4 – Solution Let represent the area of the triangle shown. Because A∆ + Asegment = Asector we see that Asegment = Asector – A∆ = 122 –  12  12 =  144 –  144 = (36 – 72) in2

Area of a Segment In Example 4, the boundaries of the segment shown are chord and minor arc . Therefore, the perimeter of the segment is given by Psegment = AB + ℓ