Lesson 9.6 Sectors and Segments pp Lesson 9.6 Sectors and Segments pp

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. 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 =, or l =. 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 =, or l =. l l c c   360 cc cc

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

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. DefinitionDefinition

sector segment

Theorem 9.21 The area of a sector is given by the proportion =, or A = A c, where A is the area of a sector, A c is the area of the circle, and  is the arc measure in degrees. Theorem 9.21 The area of a sector is given by the proportion =, or A = A c, where A is the area of a sector, A c is the area of the circle, and  is the arc measure in degrees. A A AcAc AcAc   360  

By substituting A c =  r 2 we get A = 360 r2r2 r2r2 (area of sector)

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

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

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

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

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

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

Homework pp Homework pp

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

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

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

RadiusCentral angle measureArc length (units)(degrees)(units)  /  /  / 9 RadiusCentral angle measureArc length (units)(degrees)(units)  /  /  / 9 ►A. Exercises

RadiusArc measureArea ofPerimeter (units)(degrees)the sectorof sector  /  / 3 32  / RadiusArc measureArea ofPerimeter (units)(degrees)the sectorof sector  /  / 3 32  / ►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? ►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 5

►B. Exercises 22. ►B. Exercises ° 6 6

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

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

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

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

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

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