Lesson 9 6 Sectors and Segments pp 399

Lesson 9. 6 Sectors and Segments pp. 399 -404

Objectives: 1. To derive and apply a formula for arc length. 2. To develop and apply formulas for the areas of sectors and segments. 3. To develop and apply a formula for the perimeter of a sector.

Theorem 9. 20 If the degree measure of an arc is and the circumference of the circle is c, then the length of the arc is l, given by l = , or l = c . c 360

By substituting c = 2 r we get 2 r I= 360 r = 180 (length of arc)

Definition A sector of a circle is the region bounded by two radii and the intercepted arc. A segment of a circle is the region bounded by a chord and its intercepted arc.

sector segment

Theorem 9. 21 The area of a sector is given by A the proportion = , or 360 A c A = Ac , where A is the 360 area of a sector, Ac is the area of the circle, and is the arc measure in degrees.

By substituting Ac = r 2 we get r 2 A= 360 (area of sector)

Example 1 Find the area of the segment formed by CD. Answer 1. Find the area of the sector intercepting CD. P 8 C D 60

Example 1 Find the area of the segment formed by CD. Answer 2. Find the area of equilateral ∆PDC. P 8 C D 60

Example 1 Find the area of the segment formed by CD. Answer 3. Subtract to find the area of the segment. P 8 C D 60

Example 2 Find the perimeter of the sector intercepting XY. Answer 1. Find the length of XY. X 110° U Y

Example 2 Find the perimeter of the sector intercepting XY. Answer 2. Add the two radii lengths to the arc length. X 110° U Y

Practice: Find the area of a sector of 20° in a circle of radius 6. (62)20 A= 360 720 = 360 = 2 20° 6

Homework pp. 402 -404

►A. Exercises Find the indicated measure in each diagram. The radius is r, the angle measure is , and the arc length is l. r l l= 1. 180 75° 8

►A. Exercises Find the indicated measure in each diagram. The radius is r, the angle measure is , and the arc length is l. 2. l 134° 12

►A. Exercises Find the indicated measure in each diagram. The radius is r, the angle measure is , and the arc length is l. 3. r 45° 3 2

►A. Exercises Find the indicated measure in each diagram. The radius is r, the angle measure is , and the arc length is l. 4. 15 35 4

►A. Exercises Radius Central angle measure Arc length 6. (units) (degrees) 3 50 7. 16 8. 7 9. 7 10. (units) 8 / 9 90 220 35 / 3 55 / 9

►B. Exercises Radius Arc measure Area of Perimeter (units) (degrees) the sector of sector 11. 2 55 4 16 / 15. 8 128 / 32 / +16 3 3 17. 15 13. 26 3

►B. Exercises 20. A 10 -inch pizza is cut into 8 slices. How many square inches of topping does Janelle consume if she has just one slice? 45° 5

►B. Exercises 22. 90° 6

►B. Exercises Find the area of the shaded regions. A 23. K 10 P B

■ Cumulative Review Match to each set the best description. Use each answer once. A. at least one 32. FG RS point B. at most one point C. exactly one point D. less than one point E. more than one

■ Cumulative Review Match to each set the best description. Use each answer once. 33. BC circle A A. at least one where B is on point circle A. B. at most one point C. exactly one point D. less than one point E. more than one

■ Cumulative Review Match to each set the best description. Use each answer once. A. at least one 34. DH KL point where K and L are in B. at most one point opposite half. C. exactly one planes point D. less than one point E. more than one

■ Cumulative Review Match to each set the best description. Use each answer once. A. at least one 35. AB BA point B. at most one point C. exactly one point D. less than one point E. more than one

■ Cumulative Review Match to each set the best description. Use each answer once. 36. PQRS MN A. at least one where M and point B. at most one N point are in the interior of the C. exactly one point convex D. less than one polygon point E. more than one