2002 年度中三年級
數學補充練習
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班別編號：
Mensuration
Important Terms
Capacity
Circumference
Circular Cone
Cube
Curved Surface
Cylinder
Divide
Hexagon
Lateral Face
Radius, Radii
容量
圓周
圓錐體
正方體
曲面
圓柱體
分割
六邊形
側面
半徑
Line Segment
Regular Pentagon
Right Pyramid
Perimeter
Prism
Sector
Slant Height
Sphere
Trapezium
Diameter
線段
正五邊形
直立角錐體
周界
角柱體
扇形
斜高
圓球體
梯形
直徑
Revision Notes
1.
Basic knowledge of some plane figures
Triangle
Trapezium
a
Area =
bh
2
h
(a  b)h
2
Area =
h
b
b
Circle
Area = r
Sector
2
Area =
r
360

 r 2
Perimeter = 2r +
Circumference = 2r
2.

r

360 

 2r
Basic knowledge of some solids
Cuboid
Prism
h
Volume = abh
Volume = Ah
h
Base Area
A
a
b
Cylinder
Pyramid
V
Volume = r 2 h
Curved surface area= 2rh
Total surface area = 2rh  2r 2
Mensuration_M3_02
h
1
Volume =  Base area  Height
3
r
B
A
C
N
D
P. 1
Right Circular Cone
Sphere
l 2  h2  r 2
1
Volume = r 2 h
3
Curved surface area = rl
Volume =
l
3.
r
h
Total surface area = rl  r 2
4 3
r
3
Surface area = 4r
2
r
Properties of similar plane figures and similar solids
For two similar plane figures with areas A1 , A2 and corresponding linear measures L1 , L2, we have
A1  L1 
 
A2  L2 
2
.
For two similar solids with volumes V1 ,V2 and corresponding linear measures L1 , L2, we have
V1  L1 
 
V2  L2 
3
.
Exercise
1.
The ratio of the curved surface areas of two similar cones X and Y is 16 : 25. Find the ratio of their
(a) base radii, rX : rY ,
(b) heights, hX : hY ,
(c) volumes, VX : VY ,
(d) base circumferences, CX : CY .
2.
In the figure, a sphere just fits inside a cylinder of height 2r and the cylinder just fits a cone.
(a) Find (i) radius of sphere : height of cylinder,
(ii) volume of sphere : volume of cylinder.
(b) Find, in terms of r, the volume of the space between
(i) the cylinder and the cone,
(ii) the cylinder and the sphere.
Mensuration_M3_02
P. 2
2r
3.
The figure shows a circular measuring cylinder 4cm in diameter containing water.
Three iron balls, each of diameter 2cm, are dropped into the cylinder.
What is the rise in the water level?
4.
(87ce)
A solid rectangular iron block, 4cm  2cm  1cm, is melted and recast into a cube. Find the
decrease in the total surface area. (87ce)
1cm
2cm
4cm
5.
The radii of two solid spheres made of the same material are in the ratio 2 : 3. If the smaller sphere
weighs 16kg, then find the larger one weighs. (87ce)
6.
A solid iron sphere of radius r is melted and recast into a circular cone and a circular cylinder. If both
of them have the same height h and the same base radius r, find h in terms of r. (88ce)
7.
The circular cylinder and the circular cone have the same height. The radius of the base of the
cylinder is twice of that of the cone. If the volume of the cone is
20 ㎝ 3 , what is the volume of the cylinder﹖(90ce)
Mensuration_M3_02
P. 3
8.
In the figure, the base of the conical vessel is inscribed in the
bottom of the cubical box. If the box and the conical vessel have
the same capacity, find h : r. (93ce )
h
r
9.
The figure shows a solid consisting of a cylinder of height h and a hemisphere of radius r. The area
of the curved surface of the cylinder is twice that of the hemisphere.
Find the ratio of volume of cylinder : volume of hemisphere. (93ce )
h
r
10. In the figure, A and B are two right solid cylinder with the same base radius 1. If the heights of A
and B are 1 and 2 respectively, find
the total surface area of A
.
the total surface area of B
(96ce)
B
A
1
1
11. The figure shows a frustum of a circular cone. The radii of the upper face
and the base are 1 ㎝ and 2 ㎝ respectively. If the height is 3 ㎝,
find the volume. (Leave the answer in terms of )
(96ce )
2
1
1cm
3cm
2cm
Mensuration_M3_02
P. 4
12. The figure shows a right circular cylinder with AC being a diameter of its upper face. AB and CD
are two vertical lines on the curved surface. A curve is drawn on the surface of the cylinder from B
to C. Find its shortest possible length. (Leave the answer in terms of . ) (96ce )
A
4cm
C
4cm
B
D
13. (a) Find the volume of a spherical ice ball of radius 10 cm in terms of .
(b) This ice ball is placed in an inverted conical vessel having the same radius. When the ice ball
melts, there is a 2.5% contraction in volume. The conical vessel can just hold all the water
when the ice ball melts completely. Find
(i) the volume of the water in terms of ,
(ii) the height of the conical vessel.
14. Figure (a) shows a conical container with a diameter of 10cm and a height of 12 cm. The container is
filled with water to a depth of 9cm.
10cm
(a) Find, in terms of ,
(i) the capacity of the container,
(ii) the volume of the water in the container.
12cm
9cm
Figure (a)
hcm
Figure (b)
(b) The container is then sealed at the top and is inverted. The depth of water is now h cm (see
Figure (b)). Find the value of h correct to 1 decimal place.
Mensuration_M3_02
P. 5
(Percentage Change)
15. It is given that the volume of a right circular cone is V.
Find, in terms of V, the new volume in each of the following cases:
(a) The base radius is kept constant, the height is doubled.
(b) The radius is halved, the height is doubled.
16. If the length and the width of a rectangle are decreased by 10% and 20% respectively, find the
percentage decrease in area.
17. Given a right circular cone. Its base radius is increased by 5% and its height is decreased by 10%.
Let V1 and V2 be the volumes of the original and the new cones respectively.
(a) Express V2 in terms of V1 .
(b) Find the percentage decrease in the volume of the cone.
18. A solid sphere is cut into two hemispheres. Find the percentage increase in the total surface area.
(83ce)
Mensuration_M3_02
P. 6
19. The length and width of a cuboid are each increased by 10% and the height remains unchanged. Find
the percentage increase in volume. (85ce)
20. A cylindrical hole of radius r is drilled through a solid cylinder, base radius 2r and height r, as shows
in the figure. Find the percentage increase in the total surface area. (88ce)
r
2r
r
21. The length of a rectangle is decreased by 20%. If the area remains unchanged, find the percentage
increase of its width. (96ce)
Answers
1. (a) 4:5
(b) 4:5
2.
(ii)2:3
(a)(i)1:2
240cm
3:
4
(b)(i) r 3
3
(d) 4:5
2
(ii) r 3
3
4. 4cm2
6. r
3. 1cm
5. 54kg
7.
9.
(c) 64:125
3
11. 7 cm3
Mensuration_M3_02
4000
cm 3
3
13.
(a)
14.
(a) (i) 100 cm3
(b) (i) 1300 cm 3 (ii) 39
(ii)
675
 cm3
16
(b) h = 2.0
15. (a) 2V
8. 24 : 
10. 2 : 3
(b)
17. (a) 0.99225 V1
18. 50%
20. 0%
2
12. 2   4 cm3
P. 7
1
V
2
16. 28%
(b) 0.775%
19. 21%
21. 25%