11.6 Arc Lengths and Areas of Sectors

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Presentation transcript:

11.6 Arc Lengths and Areas of Sectors Geometry 11.6 Arc Lengths and Areas of Sectors

Arc Length The length of part of the circumference. The length of the arc depends on what two things? 1) The measure of the arc. 2) The size of the circle. An arc length measures distance while the measure of an arc is in degrees.

Sector of a circle A region bounded by 2 radii and an arc. .

 Portions of a Circle: Determine the Arc measure based on the portion given. 1/3 of circumference : 6π out of a total 36π on the circle: ¼ ● 360 ½ ● 360 1/3 ● 360 1/6 ● 360

Area of a Sector Formula measure of the central angle or arc m πr2 Area of a sector = 360 The area of the entire circle! The fraction of the circle! .

m 2πr 360 . Arc Length Formula measure of the central angle or arc The circumference of the entire circle! 2πr Arc Length = 360 The fraction of the circle! .

Find the length of AB and the area of sector AOB. 1. 2. 3. 4. 5. Length of AB Area of sector AOB B 90o B 90o 240o 300o 12 120o 108o O B O A A O A A 6 12 O 2.4 O 10√2 B B Fraction of circle: Fraction of circle: Fraction of circle: Fraction of circle: Fraction of circle: ¼ 2/3 5/6 1/3 3/10 Fraction ● circumference Fraction ● circumference Fraction ● circumference ¼ ● 12π 2/3 ● 24π 5/6 ● 24π 1/3 ● 4.8π 3/10 ● 20√2π 3π units 16π units 20π units 1.6π units 6√2π units Fraction ● area Fraction ● area Fraction ● area Fraction ● area Fraction ● area ¼ ● 36π 2/3 ● 144π 5/6 ● 144π 1/3 ● 5.76π 3/10 ● 200π 9π units2 96π units2 120π units2 1.92π units2 60π units2

6. The area of sector AOB is 48π and . Find the radius of ○O. m πr2 Area of a sector = 360 270 48π = πr2 360 4 16 3 4 r2 48 = 3 4 3 r2 64 = r = 8

7. The area of sector AOB is and . Find the radius of ○O. m πr2 Area of a sector = 360 9 40 π = πr2 4 360 9 9 1 9 = r2 1 4 9 1 81 = r2 4 9 r = 2

Find the area of the shaded region Find the area of the shaded region. Point O marks the center of the circle. 8. 9. 10. 11. 60˚ 8 12 60 O 6 4 30 O O 160 π units2 9π - 18 units2 24π - 36√3 units2 8π - 8√3 units2 3

HW P. 453-454 WE #1-19

Some common fractions and measures! Arc or Central Angle Measure Fraction of the Circle 36o 108o 1/6 5/6 120o 2/3 30o 11/12 1/8 5/8 3/10 1/10 60o 300o 1/3 240o 1/12 330o 45o 225o