Immaculate Heart of Mary College
S3 Remedial Class
Lesson 16: Measuration (Sphere and Similar Solid)
Name: ________________________(
) Class: S3 (
) Date: __________
Basic Knowledge
Vocabulary List
English
中文
sphere
球體
hemisphere
半球體
surface area
表面面積
Formula
Example 1 Find the volume and surface area of a sphere of diameter 18 cm.
(Express your answers in terms of .)
Answer
4
18
Volume of the sphere  [    ( ) 3 ] cm 3
3
2
 972 cm 3
18
Surface area of the sphere  [4    ( ) 2 ] cm 2
2
 324 cm 2
1
Example 2 The volume of air inside a volleyball is 3 600 cm3 . Find the inner radius
of the volleyball. (Correct your answer to 2 significant figures.)
Answer
Let r cm be the inner radius of the volleyball.
4
  r3
3
3 600  3
r3 
4
2 700
r3 
3 600 

r  9.5 (co r r. t o 2 sig. f ig.)
∴ The inner radius of the volleyball is 9.5 cm.
Exercise 1:
1. Find the volume and surface area of a sphere of diameter 12 cm. (Express your
answers in terms of .)
2. Find the surface area of a hemisphere of radius 9 cm.
3. If the surface area of a sphere is 36 cm2 , find the radius and volume of the
sphere. (Express your answers in terms of  if necessary.)
2
4. The radii of three metallic spheres are 4 cm, 5 cm and 6 cm respectively. If they
are melted and recast into a new sphere, find the radius of the new sphere.
(Correct your answer to 1 decimal place.)
5. A sphere of radius 1.5 cm just fits into a cylindrical container of radius 1.5 cm and
height 3 cm. Find the volume of the empty space left inside the container.
(Express your answer in terms of .)
1.5 cm
3 cm
6. The figure shows a solid which is composed of a right circular cone and a
hemisphere. The base radius and height of the right circular cone are 3 cm and 4
cm respectively.
(a) Find the volume of the solid.
4 cm
(b) Find the total surface area of the solid.
(Express your answers in terms of .)
3 cm
7. The figure shows a hemispherical copper bowl of inner radius 9 cm and thickness
0.5 cm. If the density of copper is 9 g per cm3,
9 cm
0.5 cm
(a) find the total surface area of the copper bowl.
(b) find the weight of the copper bowl.
(Correct your answers to 3 significant figures.)
3
8. The figure shows a solid of height 7 cm. It is composed of a hemisphere and a
cylinder with base radius 1 cm.
(a) Find the volume of the solid.
(b) Find the total surface area of the solid.
(Express your answers in terms of .)
7 cm
1 cm
Similar Solids
Definition of similar solids:
Two solid are similar if
(1) they have the same shape.
(2) their corresponding sides are in proportion.
For example, let’s consider cones.
The volume of the cone’s formula is
height(h). Two cones are similar if
1 2
r h . So it depends on the radius(r) and the
3
r1 h1
 .
r2 h2
We can clearly say, all cubes are similar solids because the volume of cube depends
on 1 side only, as well as the spheres.
Example 3 Determine the following pairs of solids are similar or not.
(a)
(b)
4
(c)
(d)
Answer
(a) They are not similar because they do not have the same shape.
(b) They are not similar. Although they have the same shape,
r1 4
h
5
  2 and 1  , their corresponding sides are not in proportion.
r2 2
h2 3
(c)
r1 8
h
6
  2 , 1   2 , so they are similar solids.
r2 4
h2 3
(d) For spheres, because their measurations only depend on 1 parameter, radius. So
they must be similar solids.
For any two similar solids, if the ratio of their corresponding sides 
(i) the ratio of any sides 
l1
, then
l2
l1
;
l2
l
(ii) the ratio of their areas  ( 1 ) 2 ;
l2
l
(iii) the ratio of any sides  ( 1 ) 3 .
l2
5
Example 4 The height of a pyramid is 3 cm and the volume is 270 cm3 . What is the
volume of a similar pyramid of height 5 cm?
Answer
V o l u m e o f t h e la r g e p y r a mi d
5
 ( )3
V o l u m e o f t h e s m a ll p y r a mi d 3
5
∴ Volume of the large pyramid  [( )3  270] cm 3
3
 1250 cm 3
Note: We cannot use the formula directly since we do not know the length of the
radius.
Example 5 The figure shows two similar solids A and B. Each of them is composed
of a right circular cone and a hemisphere. The base radii of the cone of
solid A and solid B are 3 cm and 9 cm respectively. Find the ratio of the
volume of solid A to that of solid B.
Answer
V olu me of t h e solid A
3
 ( )3
V olu me of t h e solid B
9
1
 ( )3
3
1

27
∴ The required ratio is 1 : 27.
9 cm
3 cm
Solid A
Solid B
Exercise 2:
1. Determine the following pairs of solids are similar or not.
(a)
6
(b)
(c)
2. The figure shows two similar solids A and B.
5 cm
2 cm
A
B
(a) Find the ratio of the total surface area of solid A to that of solid B.
(b) Find the ratio of the volume of solid A to that of solid B.
3. 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
7
4. A and B are two similar bowling pins.
(a) Find the ratio of the total surface area of the small bowling pin to that of the
large one.
(b) Find the ratio of the volume of the small bowling pin to that of the large one.
50 cm
30 cm
A
B
5. A and B are two similar sectors with areas 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 cone to that of the small one.
(b) Find the ratio of the volume of the large cone to that of the small one.
B
A
6. A and B are two uniform cross-sections of two similar right prisms.
(a) Find the ratio of the total surface area of the small prism to that of the large
prism.
(b) Find the ratio of the volume of the small prism to that of the large prism.
3 cm
4 cm
A
B
8