1. Radians, Arc Length and Sector Area

2. Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of the angle formed at the centre of a circle by 2 radii which join the ends of an arc equal in length to the radius. r r x = 1 radian x = 1 rad. or 1 c

3. Radians, Arc Length and Sector Area r O 2r r 2c2c If the arc is 2r, the angle is 2 radians. Radians

4. Radians, Arc Length and Sector Area O If the arc is 3r, the angle is 3 radians. r 3r r 3c3c If the arc is 2r, the angle is 2 radians. Radians

5. Radians, Arc Length and Sector Area O If the arc is 3r, the angle is 3 radians. If the arc is 2r, the angle is 2 radians. r r If the arc is r, the angle is radians. r Radians

6. Radians, Arc Length and Sector Area O If the arc is 3r, the angle is 3 radians. r r If the arc is 2r, the angle is 2 radians. If the arc is r, the angle is radians. r Radians

7. Radians, Arc Length and Sector Area If the arc is r, the angle is radians. O r r r But, r is half the circumference of the circle so the angle is Hence, Radians

8. Radians, Arc Length and Sector Area We sometimes say the angle at the centre is subtended by the arc. Hence, r O r r x x = 1 radian Radians

9. Radians, Arc Length and Sector Area  Radians SUMMARY One radian is the size of the angle subtended by the arc of a circle equal to the radius 1 radian

10. Radians, Arc Length and Sector AreaExercises 1. Write down the equivalent number of degrees for the following number of radians: Ans: (a) (b) (c) (d) 2. Write down, as a fraction of, the number of radians equal to the following: (a) (b) (c) (d) Ans: It is very useful to memorize these conversions

11. Radians, Arc Length and Sector AreaArc Length and Sector Area Let the arc length be l. O r r l Consider a sector of a circle with angle. Then, whatever fraction is of the total angle at O,...... l is the same fraction of the circumference. So, ( In the diagram this is about one-third.) circumference

12. Radians, Arc Length and Sector Area O r r Also, the sector area A is the same fraction of the area of the circle. A circle area Arc Length and Sector Area

13. Radians, Arc Length and Sector AreaExamples 1. Find the arc length, l, and area, A, of the sector of a circle of radius 7 cm. and sector angle 2 radians. Solution: where is in radians

14. Radians, Arc Length and Sector Area 2. Find the arc length, l, and area, A, of the sector of a circle of radius 5 cm. and sector angle. Give exact answers in terms of. Solution: where is in radians rads. So, Examples

15. Radians, Arc Length and Sector Area  Radians An arc of a circle equal in length to the radius subtends an angle equal to 1 radian. 1 radian  For a sector of angle radians of a circle of radius r, the arc length, l, is given by the sector area, A, is given by SUMMARY

16. Radians, Arc Length and Sector Area 1. Find the arc length, l, and area, A, of the sector shown. O 4 cm A l 2. Find the arc length, l, and area, A, of the sector of a circle of radius 8 cm. and sector angle. Give exact answers in terms of. Exercises

17. Radians, Arc Length and Sector Area 1. Solution: O 4 cm A l Exercises

18. Radians, Arc Length and Sector Area 2. Solution: rads. So, O 8 cm A l where is in radians Exercises

19. Radians, Arc Length and Sector Area