[ In this exercise, give your answers correct to 3 significant figures if necessary. ]
1. The base of a pyramid is an isosceles right -angled triangle
where the lengths of the two equal sides are 8 cm. The height
VE of the pyramid is 15 cm. Find the volume of the pyramid.
V
15 cm
C
8 cm
E
A
8 cm
B
2. It is given that the base of a pyramid is a triangle with base
a cm and height b cm. If the height of the pyramid is h cm,
express the volume of the pyramid in terms of a, b and h.
3. In the figure, VABC is a pyramid. ABC is an isosceles
right-angled triangle. If AB  AC  40 cm and VB  VC  50 cm ,
find the volume of pyramid VABC.
V
C
A
B
4. In the figure, ABCD is a trapezium where AB  16 cm ,
AD  10 cm and CD  20 cm.
A
16 cm
B
10 cm
(a) Find the area of trapezium ABCD.
(b) If trapezium ABCD is a base of a pyramid with a height of
20 cm, find the volume of the pyramid.
D
20 cm
C
5. The height and volume of a pyramid are 12 cm and 120 cm3 respectively. Its base is a rectangle with
dimensions 6 cm  x cm. Find x.
6. The height and volume of a pyramid are 12 cm and 100 cm3 respectively. If the base of the pyramid
is a square, find the length of each side of the square base.
4.11
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7. In the figure, VABCD is a right pyramid. Its base ABCD is a
square with sides of 6 cm each. E is a point on BC such
that VE  BC and VE  10 cm . Find the total surface area of
the pyramid.
V
10 cm
C
D
A
E
6 cm B
V
8. In the figure, VABCD is a right pyramid. The base ABCD is a
square with sides of 5 cm each. The slant edge is 8 cm long.
8 cm
(a) Find the height VO of the pyramid.
D
(b) Find the volume of the pyramid.
C
O
5 cm B
A
9. The base of a right pyramid is a square with an area of 81 cm2.
The height is 15 cm. Find the length of the slant edge of the
pyramid.
10. In the figure, VABC is a pyramid. ABC is an isosceles
right-angled triangular base where AB  AC  30 cm . The
height VA of the pyramid is 20 cm.
V
30 cm
20 cm
C
(a) Find the area of VBC.
A
(b) Find the total surface area of the pyramid.
30 cm
B
11. In the figure, VABCD is a right pyramid where the base is a
rectangle with dimensions 24 cm  10 cm . The slant edge is
30 cm long.
V
30 cm
(a) Find the height VE of the pyramid.
B
(b) Find the volume of the pyramid.
(c) Find the total surface area of the pyramid.
C
E
24 cm
A
12. The figure shows a frustum with right-angled triangular bases
where AC  15 cm, AB  12 cm, PQ  12 cm and AP  10 cm .
10 cm
D
V
(a) By using similar triangles VPQ and VAC, find VA.
R
(b) By using similar triangles VPR and VAB, find PR.
(c) Hence find the volume of frustum ABCQPR.
P
Q
B
A
4.12
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C
13. In the figure, ABCDEFGH is a solid cuboid with the
height of 50 cm. Its base is a square with dimensions
20 cm  20 cm . VEFGH is a right pyramid with the same
height as the cuboid.
D
A
V
B
C
50 cm
(a) Find the total surface area of pyramid VEFGH.
E
20 cm
F
(b) If pyramid VEFGH is removed from the cuboid, find
the total surface area of the remaining solid.
G
H
20 cm
14. If a solid square-based metal pyramid is melted and recast
to form another square-based pyramid which is 21%
higher than the original pyramid, find the percentage
decrease in the length of each side of the square base.
15. The figure shows a square-based right frustum, where
AB  20 cm , PQ  12 cm and PA  8 cm.
V
(a) By using similar triangles VPQ and VAB, find VA and
the height of pyramid VABCD.
S
P
(b) Hence find the volume of frustum PQRSDABC.
R
Q
C
D
(c) Find the height of VAB from point V.
(d) Hence find the total surface area of frustum
PQRSDABC.
A
B
[ In this exercise, give your answers correct to 3 significant figures if necessary. ]
16. Find the volume of each of the following right circular cones. (Express your an swers in terms of .)
(a)
(b)
(c)
24 cm
25 cm
15 cm
9 cm
10 cm
4 cm
4.13
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17. Find the curved surface area of each of the following right circular cones. (Express your answers in
terms of .)
(a)
5 cm
(b)
0.9 m
(c)
16 cm
20 cm
24 cm
4m
18. Find the volume and total surface area of each of the following rig ht circular cones. (Express your
answers in terms of .)
(a)
(b)
(c)
16 cm
16 cm
24 cm
25 cm
17 cm
12 cm
19. The figure shows an inverted right conical paper cup. The capacity
of the paper cup is 180 cm3 and the base radius is 4 cm.
(a) Find the height of the paper cup.
(b) If the cup is filled with water, find the area of the wet surface .
20. The slant height of a right circular cone is 18 cm and the height is half of the slant height .
(a) Find the volume of the cone in terms of .
(b) Find the total surface area of the cone.
4.14
 2010 Chung Tai Educational Press. All rights reserved.
4 cm
21. The figure shows a right circular conical hat formed by rolling up a
paper sector. It is given that the slant height of the hat is 25 cm and
the perimeter of the base is 18 cm.
25 cm
(a) Find the area of the paper sector in terms of .
(b) If the cost of paper for making the hat is $10/ m 2 , find the cost
of paper for making 50 conical hats.
1
of the metal is
3
recast to form a right circular cone with the base same as the original cylinder. Find the height of
the cone.
22. (a) A metal right cylinder with both its base radius and height of 10 cm is melted.
(b) The rest of the metal is recast to form another right circular cone with the base same as the
original cylinder. Find the total surface area of this cone.
23. The figure shows an ice-cream cone where the volume of the
ice-cream is 400 cm3 . The height of the cone is 12 cm and it is
filled with ice-cream. The ratio of the volume of ice-cream outside
the cone to that inside the cone is 3 : 5 , find the base radius of the
ice-cream cone.
12 cm
24. The figure shows a chocolate in the shape of a right circular
frustum. The upper and lower base diameters are 2 cm and 3 cm
respectively.
(a) Find the volume of the chocolate.
(b) It is given that every cm 3 of chocolate weighs 3 g. How many
chocolates as shown in the figure can be produced from 1 kg of
chocolate?
2 cm
1 cm
3 cm
25. The figure shows a paper sector with an area of 120 cm2. If a right
circular cone is formed by rolling up the paper sector,
(a) find the base radius of the cone.
(b) find the height of the cone.
120cm2
(c) find the volume of the cone.
4.15
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26. A right circular frustum is formed by rotating trapezium ABCD
360 about the axis AD. It is given that AB  6 cm , AD  9 cm and
DC  15 cm .
(a) Find the volume of the frustum in terms of .
(b) Find the total surface area of the frustum.
6 cm
A
B
9 cm
D
15 cm
27. The figure shows an inverted right conical funnel with the base
radius of 5 cm and height of 16 cm. Initially, the funnel is filled
with water. After a while, the water level drops to 8 cm.
(a) Find the radius of the water surface in the figure .
C
5 cm
16 cm
(b) What percentage of water is dripped from the funnel?
8 cm
28. The figure shows an inverted right conical paper cup
containing 8 cm3 of water. The diameter of the water surface is
4 cm and the water surface is 2 cm below the rim of the cup.
2 cm
4 cm
(a) Find the depth of water.
(b) Find the area of the wet surface.
(c) Find the capacity of the cup.
29. The figure shows a rocket model made up of three parts . Solid I is a
right circular cone. Solid II is a right cylinder. Solid III is a right
circular frustum.
8 cm
15 cm
I
15 cm
II
15 cm
III
(a) Find the volume of solid III in terms of .
(b) Find the volume of the rocket model in terms of .
(c) Find the total surface area of the rocket model .
12 cm
4.16
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[ In this exercise, express your answers in terms of  if necessary. ]
30. The radius of a sphere is 1.5 cm.
(a) Find the volume of the sphere.
1.5 cm
(b) Find the surface area of the sphere.
31. The diameter of a sphere is 8 cm.
(a) Find the volume of the sphere.
(b) Find the surface area of the sphere.
8 cm
32. The diameter of a sphere is 15 m.
(a) Find the volume of the sphere.
(b) Find the surface area of the sphere.
15 m
33. Find the volume and total surface area of each of the following solids .
(a)
(b)
(c)
The diameter of the hemisphere
is 10 cm.
The circumference of the base of
the hemisphere is 20 cm.
The radius of the sphere
is 8 cm.
4.17
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34. If the volume of a sphere is 10 cm3, find the radius of the sphere. (Give your answer correct to
3 significant figures.)
35. If the volume of a sphere is 100 cm3 , find the diameter of the sphere. (Give your answer correct to
3 significant figures.)
36. If the volume of a sphere is
4
 cm3, find the surface area of the sphere.
3
37. A hemispherical pudding with the volume of 144 cm3 is
shown in the figure. Find its total surface area .
38. If the surface area of a crystal ball is 144 cm2, find the volume
of the crystal ball.
39. If the surface area of a sphere is 3 600 cm2, find the volume of
the sphere.
A
40. In the figure, O is the centre of the circle, the circumference is
36 cm. A sphere is formed by rotating the circle 360 about
diameter AOB.
(a) Find the surface area of the sphere.
O
(b) Find the volume of the sphere.
B
41. A metal hemisphere with the radius of 4 cm is melted and recast to form a metal sphere.
(a) Determine whether the total surface area of the solid increases or decreases .
(b) Find the percentage increase / percentage decrease in the total surface area of the solid. (Give your
answer correct to 3 significant figures.)
4.18
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42. If the outer diameter of a hollow metal sphere is 12 mm and the thickness is 2 mm, find the volume of
metal required to form the metal sphere.
43. A few years ago, the standard diameter of a table tennis ball for competition changed from 38 mm to
40 mm. Find the percentage increase in the surface area of a table tennis ball for competition. (Give
your answer correct to 3 significant figures.)
44. In the figure, there is a metal ball with the radius of
5 cm inside a container in the shape of right prism.
The base of the container is a rectangle of
dimensions 15 cm  12 cm . Water is poured into the
container until the metal ball is just covered by water.
(a) Find the volume of water.
12 cm
(b) Now, 10 more metal balls with diameters of 2.4 cm
each are put into the container. Assume that the
metal balls are fully immersed in water and water
does not overflow, how much does the water level
rise?
15 cm
(Give your answers correct to 3 significant figures.)
45. Figure A shows a hemisphere with the radius of r cm.
Figure B shows a solid which is formed by removing an
inverted right circular cone from a right circular
cylinder. The base radii and heights of the cone and
cylinder are all r cm.
z cm
z cm
Figure A
Figure B
(a) Show that the volumes of solids in Figures A and
B are equal.
(b) Figures C and D show the cross-sections at z cm
from the bases of Figures A and B respectively.
Show that the areas of the two cross-sections are
equal.
Figure C
Figure D
4.19
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[ In this exercise, give your answers correct to 3 significant figures if necessary. ]
46. A and B are the uniform cross-sections of two similar
prisms.
(a) Find the ratio of the total surface area of the larger
prism to that of the smaller one.
A
(b) Find the ratio of the volume of the larger prism to
that of the smaller one.
Perimeter  28 cm
47. In the figure, A and B are two similar solids. If the area
of the cross-section of solid A is 84 cm2, find the area
of the cross-section of solid B. (Express your answer in
terms of .)
B
Perimeter  21 cm
8 cm
12 cm
B
A
48. In the figure, A and B are two similar solids . If the
volume of solid B is 6 cm3 , find the volume of solid A.
10 cm
4 cm
B
A
1
of his real height is 600 cm3 . If a similar
8
bronze statue is produced with its height equals 1.5 times the real height of Bruce Lee, what is its
volume?
49. The volume of a figure of Bruce Lee with height equals
4.20
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50. (a) According to the given ratios of the volumes of the similar solids V1 : V2 , find the ratios of their
corresponding lengths 1 :  2 and the ratios of their total surface areas A1 : A2.
(i) V1 : V2  125 : 512
(ii) V1 : V2  64 : 27
(b) According to the given ratios of the total surface areas of the similar solids A1 : A2, find the ratios
of their corresponding lengths 1 :  2 and the ratios of their volumes V1 : V2 .
(i) A1 : A2  4 : 25
(ii) A1 : A2  121: 169
51. In the figure, the volume of the solid is 792 cm3 . If a similar
solid is produced such that the area of the top is 2.25 times of
the given one, find the volume of the new solid.
52. The figure shows an inverted right conical paper cup with
water. The depth of water is 5 cm. After drinking half of the
water, what is the depth of water?
5 cm
53. When a metal rod is heated, the length of the rod increases by 8% . Find the percentage increase in the
volume of the metal rod.
54. A and B are two similar sectors with the areas of 36 cm2 and
25 cm2 respectively. Two right circular cones are formed by
rolling up the two sectors.
(a) Find the ratio of the base radius of the large r cone to that
of the smaller one.
B
A
(b) Find the ratio of the volume of the larger cone to that of
the smaller one.
4.21
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55. A and B are two similar bottles with the capacities of
750 mL and 1 200 mL respectively.
(a) Find the ratio of the height of bottle A to that of
bottle B in the form of 1 : k .
(b) Find the ratio of the base area of bottle A to that of
bottle B in the form of 1 : k .
A
56. Two different sizes of ice-cream served in two similar
ice-cream cones A and B are sold in a convenience store .
The selling price of a small ice-cream is $2 and that of
the big one is $4. Given that the ratio of the heights of
the two cones is 2 : 3, which size of ice-cream is more
economical? Explain briefly.
B
A
B
57. The ratio of the radius of metal ball A to that of metal
ball B is 2 : 3. After melting the two metal balls, all the
metal is used to recast into metal ball C.
(a) Find the ratio of the volume of metal ball B to that of
metal ball C.
441
cm3, find the
(b) If the volume of metal ball B is
10
radius of metal ball C.

A
B
C
58. A cone is divided into 3 portions A, B and C by planes
parallel to the base. The ratio of the slant heights of
portions A, B and C is 1 : 2 : 1.
A
(a) Find the ratio of the curved surface areas of portions
A, B and C.
C
(b) Find the ratio of the volumes of portions A, B and C.
4.22
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B