Lesson 9.6Sectors 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 9.20If 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

= , or l = .lc

360

c360

360I = 2r

By substituting c = 2r we get

180r=

(length of arc)

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

DefinitionDefinition

sector segment

Theorem 9.21The area of a sector is given by

the proportion = , or

A = Ac , where A is the

area of a sector, Ac is the area of the circle, and is the arc measure in degrees.

AAc

360

360

By substituting Ac = r2 we get

A =360r2

(area of sector)

Example 1 Find the area of the segment formed by CD.

8

60D

C

P

Answer1. Find the area of the

sector interceptingCD.

Example 1 Find the area of the segment formed by CD.

Answer2. Find the area of

equilateral ∆PDC.8

60D

C

P

Example 1 Find the area of the segment formed by CD.

Answer3. Subtract to find the

area of the segment.8

60D

C

P

Example 2 Find the perimeter of the sector intercepting XY.

Answer1. Find the length of

XY.

X

YU

110°

Example 2 Find the perimeter of the sector intercepting XY.

Answer2. Add the two radii

lengths to the arc length.

X

YU

110°

Practice: Find the area of a sector of 20° in a circle of radius 6.

A = (62)20360

=720360

= 2

20°6

Homeworkpp. 402-404

►A. ExercisesFind the indicated measure in each diagram. The radius is r, the angle measure is , and the arc length is l.1.

875°

l l = r180

►A. ExercisesFind 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. ExercisesFind the indicated measure in each diagram. The radius is r, the angle measure is , and the arc length is l.3.

45°r

23

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

15

435

Radius Central angle measure Arc length(units) (degrees) (units)

6. 3 50

7. 16 8/9

8. 7 90

9. 7 35/3

10. 220 55/9

►A. Exercises

Radius Arc measure Area of Perimeter(units) (degrees) the sector of sector

11. 2 55

13. 4 16/3

15. 8 128/332/3+16

17. 15 26

►B. Exercises

►B. Exercises20. 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. Exercises22.

90°6

A

BP

K10

►B. ExercisesFind the area of the shaded regions.23.

■ Cumulative ReviewMatch to each set the best description. Use each answer once.

A. at least one pointB. at most one pointC. exactly one pointD. less than one pointE. more than one

point

32. FG RS

33. BC circle A where B is on circle A.

■ Cumulative ReviewMatch to each set the best description. Use each answer once.

A. at least one pointB. at most one pointC. exactly one pointD. less than one pointE. more than one

point

■ Cumulative ReviewMatch to each set the best description. Use each answer once.

A. at least one pointB. at most one pointC. exactly one pointD. less than one pointE. more than one

point

34. DH KL where K and L are in opposite half-planes

■ Cumulative ReviewMatch to each set the best description. Use each answer once.

A. at least one pointB. at most one pointC. exactly one pointD. less than one pointE. more than one

point

35. AB BA

■ Cumulative ReviewMatch to each set the best description. Use each answer once.

A. at least one pointB. at most one pointC. exactly one pointD. less than one pointE. more than one

point

36. PQRS MN where M and N are in the interior of the convex polygon