1. Radians, Arc Length and Sector Area 40: Radians, Arc Length and Sector Area

2. Radians, Arc Length and Sector AreaRadians 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 =degrees1 degree =radians

10. Radians, Arc Length and Sector AreaSUMMARY 1 radian =degrees 1 degree = radians degrees to radians  Multiply by Memory Aid Dr  by 180 Degrees to radians X by radians to degrees  Multiply by

11. 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

12. 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

13. 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

14. 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

15. 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 So, Examples Memory Aid Dr  by 180 Degrees to radians X by

16. 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

17. 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

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

19. Radians, Arc Length and Sector Area 2. Solution: So, O 8 cm A l where is in radians Exercises Memory Aid Dr  by 180 Degrees to radians X by

20. Radians, Arc Length and Sector Area

21. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

22. 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 r O r r x

23. Radians, Arc Length and Sector Area Arc 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

24. Radians, Arc Length and Sector Area 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

25. Radians, Arc Length and Sector Area SUMMARY  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