Worksheet -

May’ 23

Further Pure Mathematics

Topic: Circular Measure

Prepared by Faisal Mizan

Senior Teacher of Mastermind
 Pure Mathematics

1. Convert the following angles from degrees to radians. (Give your answers in terms of 𝜋)

(a) 15° (b) 18° (c) 45° (d) 75° (e) 225°

2. Convert the following angles from radians to degrees. (Give your answers correct to 1 d.p. where
necessary.)
3𝜋 5𝜋 6𝜋
(a) (b) (c) (d) 3.2 (e) 2.56
4 6 7

3. Convert the following angles from degrees to radians, giving your answers correct to 3 significant figures:

(a) 25° (b) 37.4° (c) 78.9° (d) 142° (e) 308°

4. Find in terms of 𝜋, the arc length of a sector of:

𝜋 2𝜋
(a) Radius 6 cm and angle 4 (b) Radius 5 cm and angle 5

3𝜋 5𝜋
(c) Radius 10 cm and angle 8
(d) Radius 18 cm and angle 6

5. Find the arc length of a sector of:

(a) Radius 8 cm and angle 1.2 radian (b) Radius 2.5 cm and angle 0.8 radian

6. Find in radians, the angle of a sector of:

(a) Radius 4 cm and arc length 5 cm (b) Radius 9 cm and arc length 13.5 cm

7. Find the perimeter of each of these sectors.

(a) (b) (c)

8. Find in term of 𝜋, the area of a sector of:

𝜋 3𝜋
(a) Radius 6 cm and angle 3 (b) Radius 15 cm and angle 5

7𝜋 5𝜋
(c) Radius 10 cm and angle 10 (d) Radius 9 cm and angle 6

9. Find the area of sector of:

(a) Radius 4 cm and angle 1.3 radian (b) Radius 3.8 cm and angle 0.6 radian

10. Find, in radians, the angle of a sector of:

(a) Radius 3 cm and area 5 cm2 (b) Radius 7 cm and area 30 cm2

11. 𝑃𝑂𝑄 is the sector of a circle, center 𝑂, radius 10 cm. The length of arc 𝑃𝑄 is 8 cm. Find:

(a) angle 𝑃𝑂𝑄, in radians (b) The area of the sector 𝑃𝑂𝑄.

FAISAL MIZAN 1
 Pure Mathematics

12. A sector of a circle, radius 𝑟 cm, has a perimeter of 150 cm. Find an expression, in term of 𝑟,

for the area of the sector.

13. The points 𝑃 and 𝑄 lie on the circumference of a circle with centre 𝑂 and radius 𝑟 cm.

Angle 𝑃𝑂𝑄 = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠. The segment shaded in Figure has area 𝐴 cm2
1
Show that 𝐴 = 2 𝑟 2 (𝜃 − sin 𝜃)

14. Figure shows a sector of a circle of radius 10 cm and centre 𝑂. The area of triangle 𝑂𝐴𝐵 is 20 cm2 and

The size of angle 𝐴𝑂𝐵 is 𝜃 radians. Find, to 3 significant figures,

(a) the value of 𝜃, (b) the length of the arc 𝐴𝐵, (c) the area of the shaded segment.

15. Figure shows sector 𝐴𝑂𝐵 of a circle with centre 𝑂 and radius 𝑟cm.

The angle 𝐴𝑂𝐵 is 1.5 radians and the length of arc 𝐴𝐵 is 12 cm. Calculate:

(a) the value of 𝑟, (b) the area of the sector 𝐴𝑂𝐵.

16. Figure shows the sector 𝐴𝑂𝐵 of a circle with centre 𝑂 and radius 12 cm.

The angle 𝐴𝑂𝐵 is 𝜃 radians and the area of the sector is 192 cm2. Calculate:

(a) the value of 𝜃, (b) the length, in cm, of the arc 𝐴𝐵.

17. The diagram shows a sector 𝐴𝑂𝐵 of a circle, centre 𝑂, radius 15 cm.

The length of the arc 𝐴𝐵 is 12 cm.

(a) Find, in radians, angle 𝐴𝑂𝐵. (b) Find the area of the sector 𝐴𝑂𝐵.

18. Figure shows a sector 𝑂𝐴𝐵 of a circle. The circle has centre 𝑂 and radius 10 cm.

The area of the sector is 25 cm2 and angle 𝐴𝑂𝐵 = 𝜃 radians. Find:

(a) the value of 𝜃, (b) the length of the arc 𝐴𝐵.

19. Figure shows a sector 𝑂𝐴𝐵 of a circle, centre 𝑂.

The area of the sector is 27 cm2.

The size of angle 𝐴𝑂𝐵 is 1.5 radians. Find the perimeter of the sector.

20. Figure shows the sector 𝐴𝑂𝐵 of a circle with centre 𝑂.

The radius of the circle is 13cm and angle 𝐴𝑂𝐵 = 2 radians.

(a) Find the length of the arc 𝐴𝐵. (b) Find the area of the sector 𝐴𝑂𝐵.

21. Figure shows a sector 𝑂𝐴𝐵 & 𝑂𝐷𝐶 of a circle where angle 𝐴𝑂𝐵 = 𝜃 radians.

Given that 𝑂𝐵 = 15cm, 𝑂𝐷 = 10cm

The area of the region 𝐴𝐵𝐶𝐷, shown shaded in figure, is 100 cm2. Find:

FAISAL MIZAN 2
 Pure Mathematics

(a) the value of 𝜃, (b) the perimeter of the region 𝐴𝐵𝐷𝐶.

22. The diagram shows the right-angled triangle 𝑂𝐴𝐵.

𝜋
The point 𝐶 lies on the line 𝑂𝐵. Angle 𝑂𝐴𝐵 = 2
radians and angle 𝐴𝑂𝐵 = 𝜃 radians.

𝐴𝐶 is an arc of the circle, centre 𝑂, radius 12 cm and 𝐴𝐶 has length 9.6 cm.

(a) Find the value of 𝜃. (b) Find 𝐴𝐵

(c) Find the area of the shaded region. (d) Find the perimeter of the shaded region.

23. The diagram shows a circle, centre 𝑂, radius 8 cm. Points 𝑃 and 𝑄 lie on the circle

such that the chord 𝑃𝑄 = 12 cm and angle 𝑃𝑂𝑄 = 𝜃 radians.

(a) Show that 𝜃 = 1.696, correct to 3 decimal places.

(b) Find the perimeter of the shaded region.(c) Find the area of the shaded region.

24. Figure shows a sector of a circle. The circle has radius 𝑟 cm and

the sector has angle 𝜃 radians. The sector has an arc length of 18𝜋 cm

and an area of 126𝜋 cm2. Find: (i) the value of 𝑟, (ii) the exact value of 𝜃.

25. The sector 𝑂𝐴𝐵 of a circle, centre 𝑂, has area 48 cm2. The length of the arc 𝐴𝐵 is 8 cm and the size of

angle 𝐴𝑂𝐵 is 𝜃 radians. Find: (i) the radius of sector 𝑂𝐴𝐵, (ii) the value of 𝜃.

26. The figure shows a circle, centre 𝑂, with radius 10 cm. The lines 𝑋𝐴 and 𝑋𝐵
2𝜋
are tangents to the circle at 𝐴 and 𝐵 respectively, and angle 𝐴𝑂𝐵 is 3
radians.

(a) Find the perimeter of the shaded region. (b) Find the area of the shaded region.

27. The diagram shows a circle, centre 𝑂, radius 8 cm. The points 𝑃 and 𝑄 lie on the circle. The lines 𝑃𝑇 and
3𝜋
𝑄𝑇 are tangents to the circle and ∠𝑃𝑂𝑄 = 4
radians.

(i) Find the length of 𝑃𝑇. (ii) Find the area of the shaded region.

(iii) Find the perimeter of the shaded region.

28. The points 𝐴, 𝐵 and 𝐶 lie on a circle centre 𝑂, radius 6 cm. The tangents to the circle at 𝐴 and 𝐶 meet at

the point 𝑇. The length of 𝑂𝑇 is 10 cm. Find

(i) the ∠𝑇𝑂𝐴 in radians, (ii) the area of the region 𝑇𝐴𝐵𝐶𝑇,

(iii) the perimeter of the region 𝑇𝐴𝐵𝐶𝑇.

FAISAL MIZAN 3
 Pure Mathematics

29. The circles with centres 𝐶1 and 𝐶2 have equal radii of length 𝑟 cm.

The line 𝐶1 𝐶2 is a radius of both circles. The two circles intersect at 𝐴 and 𝐵.

(a) Given that the perimeter of the shaded region is 4𝜋 cm, find the value of 𝑟

(b) Find the exact area of the shaded region.

30. Figure shows a sector 𝐴𝑂𝐵 of a circle with centre 𝑂 and

radius 𝑟cm and a triangle 𝐵𝑂𝐶. The angle of sector 𝐴𝑂𝐵 is

0.8 radians. The points 𝑂, 𝐴 and 𝐶 lie on a straight line so

that 𝐶𝐵 is the tangent to the circleat 𝐵. Given that the

area of the shaded region in Figure 1 is 101cm2, find the value of 𝑟. Give your answer correct to 3 significant

figures. [Jan 20/P1/Q4]

31. The diagram shows a right-angled triangle 𝑂𝑃𝑄 and a circle,

centre 𝑂 and radius 𝑟 cm, which cuts 𝑂𝑃 and 𝑂𝑄 at 𝐴 and 𝐵 respectively.

Given that 𝐴𝑃 = 6 cm, 𝑃𝑄 = 5 cm, 𝑄𝐵 = 7 cm and angle 𝑂𝑃𝑄 = 90°,

Find: (a) the length of the arc 𝐴𝐵, (b) the area of the shaded region.

32. Figure shows the sector, 𝐴𝑂𝐵 of a circle with centre 𝑂

and radius 8 cm. A circle of radius 2 cm touches the lines 𝑂𝐴

𝜋
and 𝑂𝐵 and the arc 𝐴𝐵. Angle 𝐴𝑂𝐵 is 2 radians, 0 < 𝜃 < 4 .

(a) Find, to 4 significant figures, the value of 𝜃.

(b) Find, to 3 significant figures, the area of the region shaded in Figure. [Jan 13/P2/Q1]

Answers:
𝜋 𝜋 𝜋 5𝜋 5𝜋
[1] (a) , (b) , (c) , (d) (e) [2] (a) 135°, (b) 150°, (c) 154.3°, (d) 183.3°, (e) 146.7° [3] (a) 0.436, (b) 0.653,
12 10 4 12 4

3𝜋 15𝜋
(c) 1.38, (d) 2.48, (e) 5.38, [4] (a) cm, (b) 2𝜋 cm, (c) cm, (d) 15𝜋 cm, [5] (a) 9.6 cm, (b) 2 cm,
2 4

135
[6] (a) 1.25 rad, (b) 1.5 rad, [7] (a) 12.4 cm, (b) 32 cm, (c) 31 cm, [8] (a) 6𝜋 cm2, (b) 2
𝜋 cm2, (c) 35𝜋 cm2,

135
(d) 4
𝜋 cm2, [9] (a) 10.4 cm2, (b) 4.332 cm2, [10] (a) 1.11 rad, (b) 1.22 rad, [11] (a) 0.8 rad, (b) 40 cm2,

8 1
[12] 𝑟(75 − 𝑟), [14] (a) 0.412, (b) 4.12, (c) 0.576, [15] (a) 8, (b) 48, [16] (a) 3, (b) 32, [17](a) 0.8, (b) 90, [18] (a) 2,

8
(b) 5, [19] 21, [20] (a) 26, (b) 169, [21] (a) 5, (b) 50, [22] (a) 0.8, (b) 12.4 (c) 16.8 (d) 27.3, [23](b) 48.7, (c) 179,

9 2
[24] (i) 14, (ii) 7 𝜋, [25](i) 12,(ii) 3, [26] (a) 55.6, (b) 68.5, [27] (i) 19.3, (ii) 79.1, [28] (i) 0.927, (ii) 128, (iii) 42.6,

9
[29] (a) 𝑟 = 3, (b) 6𝜋 − 2 √3, [30] 29.7, [31] (a) 2.37, (b) 22.9, [32] (a) 0.3398, (b) 9.18,

FAISAL MIZAN 4