1. 12 Area and Volume (II)
12.1 Circumferences and Areas of Circles
12.2 Arcs and Sectors
12.3 Volumes and Total Surface Areas of Cylinders

2. 12.1 Circumferences and Areas of Circles
A. Circumferences of Circles

3. Example 1T Solution: 12 Area and Volume (II)
The circumference of a clock is 50 cm. Find its radius. (Give the answer correct to 1 decimal place.)
Solution:
Let r cm be the radius of the clock.
Circumference  2pr
50 = 2pr
r  8.0 (cor. to 1 d. p.)
∴ The radius of the clock is 8.0 cm.

4. Example 2T Solution: 12 Area and Volume (II)
In the figure, O is the centre of the larger semicircle. Find the perimeter of the shaded region. (Take p  3.14.)
Solution:
Half of the circumference of the larger circle
 cm
Half of the circumference of each smaller circle
 cm
∴ Perimeter of the shaded region  (  2  12.56) cm
 cm

5. Example 3T Solution: 12 Area and Volume (II)
The diameter of a coin is 1.4 cm. If it rolls a distance of 79.2 cm, how many turns does it make?
(Take p  )
Solution:
When the coin makes one turn, the distance covered by the coin equals the circumference of the coin.
Distance covered in one turn
Number of turns
∴ The coin makes 18 turns.

6. 12.1 Circumferences and Areas of Circles
B. Areas of Circles

7. Example 4T Solution: 12 Area and Volume (II)
The area of a circle is 250 mm2. Find the radius of the circle.
 (Give the answer correct to 1 decimal place.)
Solution:
Let r mm be the radius of the circle.
Area  pr2
r  8.9 (cor. to 1 d. p.)
∴ The radius of the circle is 8.9 mm.

8. Example 5T Solution: 12 Area and Volume (II)
A student drew a logo which consists of one larger semicircle and two smaller semicircles. Find the area of the logo.
(Give the answer correct to 1 decimal place.)
Solution:
Area of the logo

9. Example 6T Solution: 12 Area and Volume (II)
In the figure, OAB is a quadrant of a circle.
O is the centre and OB is the radius.
(a) Find the area of the coloured region.
(b) Find the perimeter of the coloured region.
(Give the answers correct to 3 significant figures.)
Solution:
(a) Area of the coloured region

10. Example 6T Solution: 12 Area and Volume (II)
In the figure, OAB is a quadrant of a circle.
O is the centre and OB is the radius.
(a) Find the area of the coloured region.
(b) Find the perimeter of the coloured region.
(Give the answers correct to 3 significant figures.)
Solution:
(b)
Perimeter of the coloured region

11. 12.1 Circumferences and Areas of Circles
C. Other Problems about Circumferences and Areas of Circles

12. Example 7T Solution: 12 Area and Volume (II)
If the ratio of the areas of two circles is 4 : 25 and the diameter of the smaller circle is 4 cm, find the area of the larger circle. (Give the answer correct to 1 decimal place.)
Solution:
Let 4k cm2 and 25k cm2 be the areas of the two circles respectively, where k is a non-zero constant
 Area of the larger circle

13. Example 8T Solution: 12 Area and Volume (II)
The circumference of a circle is measured as 95 cm correct to the nearest cm.
(a) Find the least possible radius of the circle.
(b) Find the least possible area of the circle.
(Give the answers correct to 2 significant figures.)
Solution:
(a)
Maximum absolute error
Lower limit of the radius of the circle
∴ Least possible circumference of the circle
(cor. to 2 sig. fig.)

14. Example 8T Solution: 12 Area and Volume (II)
The circumference of a circle is measured as 95 cm correct to the nearest cm.
(a) Find the least possible radius of the circle.
(b) Find the least possible area of the circle.
(Give the answers correct to 2 significant figures.)
Solution:
(b) Least possible area of the circle
(cor. to 2 sig. fig.)

15. 12.2 Arcs and Sectors

16. 12.2 Arcs and Sectors

17. 12.2 Arcs and Sectors
A. Arc Lengths

18. Example 9T Solution: 12 Area and Volume (II)
In the figure, O is the centre of the circle.
OA  9 cm and AOB  140. Find the major arc AB.
(Give the answer correct to 3 significant figures.)
Solution:
Major arc AB
(cor. to 3 sig. fig.)

19. Example 10T Solution: 12 Area and Volume (II)
In the figure, major arc AB = 22p cm and OA  15 cm.
Find the angle at the centre θ.
Solution:
Major arc AB

20. 12.2 Arcs and Sectors
B. Areas of Sectors

21. Example 11T Solution: 12 Area and Volume (II)
The figure shows a sector AOB. OA  8 cm and
AOB  110. Find the area of the sector.
(Give the answer correct to 3 significant figures.)
Solution:
Area of sector AOB
(cor. to 3 sig. fig.)

22. Example 12T Solution: 12 Area and Volume (II)
In the figure, XOY is a sector. XOY  150 and the area of sector XOY is 15p cm2. Find the radius OX.
Solution:
Let r cm be the radius OX.
 The radius OX is 6 cm.

23. Example 13T Solution: 12 Area and Volume (II)
The figure shows the design of a golden earring.
(a) Find the area of the earring.
(b) If the cost of gold is $300/cm2, find
the cost of making the earring.
(Give the answers correct to 3 significant figures.)
Solution:
(a) Area of the earring
(b) Cost of making the earring
(cor. to 3 sig. fig.)
(cor. to 3 sig. fig.)

24. 12.3 Volumes and Total Surface Areas of Cylinders
A. Volumes of Cylinders

25. Example 14T Solution: 12 Area and Volume (II)
The figure shows a cylinder with a base radius of 6 cm and a height of 10 cm. Find the volume of the cylinder. (Give the answer correct to 3 significant figures.)
Solution:
Volume of the cylinder
(cor. to 3 sig. fig.)

26. Example 15T Solution: 12 Area and Volume (II)
The volume of a cylinder is 300 cm3. If its height is 10 cm, find its base radius.
(Give the answer correct to 1 decimal place.)
Solution:
Let r cm be the base radius of the cylinder.
Volume of the cylinder
(cor. to 1 d. p.)
 The base radius of the cylinder is 3.1 cm.

27. Example 16T Solution: 12 Area and Volume (II)
The figure shows a water pipe that is 4 m long. The outer radius
and the thickness are 5 cm and 0.5 cm respectively.
(a) Find the volume of material required to make the pipe.
(b) If the material costs $0.05/cm3, find the cost of making the pipe.
(Give the answers correct to 1 decimal place.)
Solution:
(a) Volume of material required
 Volume of the whole cylinder  Volume of the space
(b) Cost of making the pipe
(cor. to 1 d. p.)
(cor. to 1 d. p.)

28. Example 17T Solution: 12 Area and Volume (II)
10 plastic cubes each with a volume of 1 cm3 are put into a cylindrical cup. There is some water in the cup. The ratio of the side of each plastic cube to the base radius of the cup is 1 : 3. If the cubes are totally covered by the water, find the rise in the water level. (Give the answer correct to 2 decimal place.)
Solution:
Base radius of the cup
= 3  Side of a plastic cube
Volume of 10 plastic cubes
Let h cm be the rise in the water level.
(cor. to 2 d. p.)
 The rise in the water level is 0.35 cm.

29. 12.3 Volumes and Total Surface Areas of Cylinders
B. Total Surface Areas of Cylinders

30. Example 18T Solution: 12 Area and Volume (II)
The figure shows a cylinder with a base diameter of 10 cm and a height of 7 cm. Find the total surface area of the cylinder.
(Give the answer correct to 1 decimal place.)
Solution:
Total surface area
(cor. to 1 d. p.)

31. Example 19T Solution: 12 Area and Volume (II)
In the figure, find the total surface area of the solid.
(Give the answer correct to 1 decimal place.)
Solution:
Divide the solid into two parts as shown in the figure.
Surface area of the upper part
 Area of the top circle + Area of the lateral face

32. Example 19T Solution: 12 Area and Volume (II)
In the figure, find the total surface area of the solid.
(Give the answer correct to 1 decimal place.)
Solution:
Surface area of the lower part
 Total surface area of the cylinder
– Area of the smaller circle on the top
 Total surface area of the solid
(cor. to 1 d. p.)

33. Follow-up 1 Solution: 12 Area and Volume (II)
The circumference of a postmark is 12 cm. Find its diameter. (Give the answer correct to 1 decimal place.)
Solution:
Let d cm be the diameter of the postmark.
Circumference  pd
12  p  d
d  3.8 (cor. to 1 d. p.)
∴ The diameter of the postmark is 3.8 cm.

34. Follow-up 2 Solution: 12 Area and Volume (II)
In the figure, O is the common centre of the two semicircles. Find the perimeter of the coloured region. (Take p  3.14.)
Solution:
Half of the circumference of the larger circle
 cm
Half of the circumference of the smaller circle
 6.28 cm
∴ Perimeter of the coloured region  (18.84  6.28  4  4) cm
 cm

35. Follow-up 3 Solution: 12 Area and Volume (II)
In a circus, a clown is standing on a wheel. If the wheel covers a distance of 440 cm in 5 revolutions, what is the
radius of the wheel? (Take p  )
Solution:
When the wheel makes one revolution, the distance covered by the wheel equals the circumference of the wheel.
Let r cm be the radius of the wheel.
∴ The radius of the wheel is 14 cm.

36. Follow-up 4 Solution: 12 Area and Volume (II)
The area of a circle is 700 cm2. Find the diameter of the circle.
 (Give the answer correct to 1 decimal place.)
Solution:
Let r cm be the radius of the circle.
Area  pr 2
700  pr 2
r 
r 
∴ Diameter  2  cm

37. Follow-up 5 Solution: 12 Area and Volume (II)
The figure shows the floor plan of a room which consists
of a square and two semicircles. Find the area of the room.
(Give the answer correct to 1 decimal place.)
Solution:
Area of the room
 Area of a circle with a diameter of 16 m  Area of the square 
 m2 (cor. to 1 d. p.)

38. Follow-up 6 Solution: 12 Area and Volume (II)
In the figure, OPQ is a quadrant of a circle with centre O. OP and OQ are the radii. OST is an isosceles triangle.
(a) Find the area of the coloured region.
(b) Find the perimeter of the coloured region.
(Give the answers correct to 3 significant figures.)
Solution:
(a)
Area of the coloured region
  Area of the circle – Area of OST

39. Follow-up 6 Solution: 12 Area and Volume (II)
In the figure, OPQ is a quadrant of a circle with centre O. OP and OQ are the radii. OST is an isosceles triangle.
(a) Find the area of the coloured region.
(b) Find the perimeter of the coloured region.
(Give the answers correct to 3 significant figures.)
Solution:
(b)
ST = cm =
(Pyth. theorem)
Perimeter of the coloured region
  Circumference + PS + ST + TQ

40. Follow-up 7 Solution: 12 Area and Volume (II)
The ratio of the diameters of two circles is 3 : 4 and the circumference of the larger circle is 64p cm.
(a) Find the radius of the smaller circle.
(b) Find the circumference of the smaller circle.
Express the answer in terms of p.
Solution:
(a)
Let 3k cm and 4k cm be the diameters of the two circles respectively, where k is a non-zero constant.
(b)
Circumference of the smaller circle ∴

41. Follow-up 8 Solution: 12 Area and Volume (II)
The radius of a circle is measured as 10 mm correct to the nearest mm.
(a) Find the largest possible circumference of the circle in terms of p.
(b) Find the range of the actual circumference of the circle in terms of p.
Solution:
(a)
Maximum absolute error
Upper limit of the radius of the circle
∴ Largest possible circumference of the circle

42. Follow-up 8 Solution: 12 Area and Volume (II)
The radius of a circle is measured as 10 mm correct to the nearest mm.
(a) Find the largest possible circumference of the circle in terms of p.
(b) Find the range of the actual circumference of the circle in terms of p.
Solution:
(b)
Lower limit of the radius of the circle
∴ Least possible circumference of the circle
∴ The actual circumference of the circle
lies between 19p mm and 21p mm.

43. Follow-up 9 ( Solution: ( 12 Area and Volume (II)
In the figure, O is the centre of the circle.
OA  12 cm and AOB  120. Find AB.
Express the answer in terms of p. (
Solution: ( cm

44. Follow-up 10 ( Solution: ( 12 Area and Volume (II)
In the figure, AB = 7p m and radius OA  10 m.
Find the angle at the centre θ. (
Solution: (

45. Follow-up 11 Solution: 12 Area and Volume (II)
The figure shows a sector AOB. OA  9 cm and
AOB  70. Find the area of the sector.
(Give the answer correct to 3 significant figures.)
Solution:
Area of sector AOB
(cor. to 3 sig. fig.)

46. Follow-up 12 Solution: 12 Area and Volume (II)
In the figure, POQ is a sector. POQ  36 and the area of sector POQ is 25.6p cm2. Find the radius OP.
Solution:
Let r cm be the radius OP.
 The radius OP is 16 cm.

47. Follow-up 13 Solution: 12 Area and Volume (II)
The figure shows a magnetic letter ‘C’.
(a) Find the area of the magnetic letter.
(b) If the cost of a magnet is $0.5/cm2, find the cost of
making the magnetic letter.
(Give the answer correct to 3 significant figures.)
Solution:
(a) Area of the magnetic letter
(b) Cost of making the magnetic letter
(cor. to 3 sig. fig.)
(cor. to 3 sig. fig.)

48. Follow-up 14 Solution: 12 Area and Volume (II)
The figure shows a cylinder with a base radius of 5 cm and a height of 7 cm. Find the volume of the cylinder. (Give the answer correct to 3 significant figures.)
Solution:
Volume of the cylinder
(cor. to 3 sig. fig.)

49. Follow-up 15 Solution: 12 Area and Volume (II)
The height and volume of a circular coin are 2 mm and 1413 mm3 respectively. Find the base diameter of the coin. (Give the answer correct to 1 decimal place.)
Solution:
Let d mm be the base diameter of the coin.
Volume of the coin
(cor. to 1 d. p.)
 The base diameter of the coin is 30.0 mm.

50. Follow-up 16 Solution: 12 Area and Volume (II)
The figure shows a pipe which is 1 m long. Its outer radius
is 5 cm and its thickness is 0.3 cm.
(a) Find the volume of material required to make the pipe.
(b) If the material cost is $0.05/cm3, find the total cost of making the pipe.
(Give the answers correct to 1 decimal place.)
Solution:
(a) Volume of material required
 Volume of the whole cylinder – Volume of the space
(b) Total cost of making the pipe
(cor. to 1 d. p.)
(cor. to 1 d. p.)

51. Follow-up 17 Solution: 12 Area and Volume (II)
In the figure, the base radius of a cylindrical glass is 4 cm. 5 identical coins are put into the glass with water.
(a) If the ratio of the base radius of each coin to the base radius of the glass is 1 : 2 and the thickness of each coin is 0.4 cm, find the volume of a coin. Express the answer in terms of .
Solution:
(a) Base radius of each coin
 Base radius of the cylindrical glass
 Volume of a coin

52. Follow-up 17 Solution: 12 Area and Volume (II)
In the figure, the base radius of a cylindrical glass is 4 cm. 5 identical coins are put into the glass with water.
(a) If the ratio of the base radius of each coin to the base radius of the glass is 1 : 2 and the thickness of each coin is 0.4 cm, find the volume of a coin. Express the answer in terms of .
(b) Find the rise in the water level after the coins are put into the glass.
Solution:
(b) Volume of 5 coins
Let h cm be the rise in the water level.
 The rise in the water level is 0.5 cm.

53. Follow-up 18 Solution: 12 Area and Volume (II)
The figure shows a cylinder with a base radius of 7 cm and a height of 10 cm. Find the total surface area of the cylinder. (Give the answer correct to 1 decimal place.)
Solution:
Total surface area
(cor. to 1 d. p.)

54. Follow-up 19 Solution: 12 Area and Volume (II)
In the figure, find the total surface area of the solid.
(Give the answer correct to 1 decimal place.)
Solution:
Divide the solid into two parts as shown in the figure.
Surface area of the upper part
 Area of the top circle
+ Area of the lateral face

55. 12 Area and Volume (II) Surface area of the lower part
 Total surface area of the cylinder – Area of the smaller circle on the top
 Total surface area of the solid
(cor. to 1 d. p.)