Volume of Solids 1

VOLUME OF SOLIDS

2.1 Cubes

Volume = length× length× length

= l3

Example

Volume = 10 × 10 × 10

= 1000 unit3

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

10

10

10

5

5

5

15

15

15

20

20

20

Volume of Solids 2

2.2 Cuboids

Volume = length x breadth x height

Example

unit 24324 Volume 3=××=

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

10

5

3

12

8

4

15

10

5

4

3

2

Volume of Solids 3

2.3 Right Prisms

Volume of a right prism = area of cross-section x height (length)

Example

Volume = 2

1×5 ×4×12

= 120 unit3

Exercise 1

Volume =

EXERCISE 2

Volume =

EXERCISE 3

Volume =

12

5

4

6

4

3

5

6

7

10 8

9

Volume of Solids 4

2.4. Right Circular Cylinders

Volume = Area of circle × Height

= π r2

h

Example

Volume = 7

22 x 14 x 14 x 10 = 6160 unit

3

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

10

7

10

14

21

10

14

20

Volume of Solids 5

2.5 Right Pyramids

Volume = ×3

1base area× height

Example

Volume = ×3

1(6×8) 4×

= 64 unit3

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

4

6

8

3

8

8

12

6

8

9

10

8

Volume of Solids 6

2.6 Right Circular Cones

Volume = 3

1 x base area x height

=3

1πr

2h

Example

Volume = 3

1 x

7

22 x 3 x 3 x 7 = 66 unit

3

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

7

3

6

7

12

6

15

3

Volume of Solids 7

2.7 Spheres

Volume = 3

4Лr

3

Example

Volume = 3

4 x

7

22 x 7 x 7 x 7

= 1437.33 unit3

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

Volume of Solids 8

2.8 Hemispheres (half of a sphere)

Volume = 3

2Лr

3

Example

Volume = 3

2 x

7

22 x 7 x 7 x 7

= 718.667 unit3

Exercise 1

Volume =

Exercise 2

Volume =

Exercise 3

Volume =

7/2

7 14

21

Volume of Solids 9

2.9 Questions Base On Examination Format.

DIAGRAM 1

Example 1 Diagram 1 shows a solid cone with base radius of 5 cm and height of 3 cm. A

small hemisphere with radius 1 cm is carved out of the solid. Find the volume,

in cm3, of the remaining solid. Use

7

22=π .

Volume of the cone = 3

1 x

7

22 x 5 x 5 x3 = 78.57 cm

3

Volume of the hemisphere = 3

2 x

7

22 x 1 x 1 x 1 = 2.10 cm

3

The volume of the remaining solid = 78.57 cm3 - 2.10 cm

3 = 76.47 cm

3

3cm

1 cm 5 cm

Volume of Solids 10

DIAGRAM 2

2 Diagram 2 shows the tip of a cone touches the top of the cuboid and the base rests on the

base of the cuboid. If the cone is taken out of the solid. Calculate the volume, in cm3,

of the remaining solid. Use 7

22=π .

DIAGRAM 3

3 Diagram 3 shows a hemisphere resting on top of a cylinder, both having bases of

identical area. The height of cylinder is 10 cm and the diameter of the cylinder is 7 cm.

Find the volume, in cm3, of the composite object. Use

7

22=π .

5 cm

5 cm

10 cm

10

7

Volume of Solids 11

4 In Diagram 4, two identical cones fit exactly on top of each other in a cylinder.

Diagram 4

The volume of each cone is 132 cm³. If the area of the base of the cone is 9π cm², find the

volume of the cylinder. Use 7

22=π .

DIAGRAM 5

5 The diagram 5 shows a right prism with a hollow cylinder.If the diameter of the hollow

cylinder is 3.5 cm, find the volume of the solid. (Use 7

22=π ).

8 cm

5cm

6 cm

Volume of Solids 12

Diagram 6

6 The solid as shown in the diagram 6 is made up of a cylinder and a cone. Calculate the

volume of the solid. (Use 7

22=π ).

Diagram 7

7 Diagram 7 shows a solid formed by combining a right prism with a half cylinder on the

rectangular ABCD.

BF = CE = 10 cm , FG = EH = 8 cm and BC = 13cm.

Calculate the volume,in cm3, of the solid.

[use 7

22=π ]

15 cm

9 cm

6 cm

Volume of Solids 13

Diagram 8

8 Diagram 8 shows a solid formed by combining a cone with a hemisphere.Find the

volume of the composite in cm3.

[use 7

22=π ] .

9 Diagram 9 shows a solid cuboid . A cylinder with radius 4 cm and height 7 cm is taken

out of the solid. Calculate the volume, in cm3, of the remaining solid.[use

7

22=π ] .

13 cm

5 cm

.

10 cm

15 cm

12 cm

Diagram 9

Volume of Solids 14

12 cm

10 cm

3 cm

12 cm

= =

= =

Diagram 10

10 Diagram 10 shows a solid formed by combining a right pyramid with a cuboid . Calculate

the volume,in cm3, of the solid.

[use 7

22=π ]

2.10 Past Year SPM Questions

Nov 2003, Q6

1. Diagram shows a solid formed by combining a right pyramid with a half cylinder on the

rectangular plane DEFG.

DE = 7 cm, EF = 10 cm and the height of the pyramid is 9 cm.

Calculate the volume, in cm3, of the solid.

[Use 7

22=π ] [4 marks]

Volume of Solids 15

July 2004, Q2 2. Diagram 1 shows a solid formed by joining a right prism and a right pyramid.

Right angled triangle PST is the uniform cross-section of the prism. PQRS is a square and the

height of the pyramid is 7 cm.

Calculate the value, in cm3, of the solid. [4 marks]

Nov 2004, Q2

3. Diagram shows a solid formed by joining a cone and a cylinder.

The diameter of the cylinder and the diameter of the base of the cone are both 7 cm. The volume

of the solid is 231 cm3.

By using 7

22=π , calculate the height, in cm

3, of the cone. [4 marks]

Volume of Solids 16

July 2005, Q8 4. Diagram 4 shows a container formed by combining a half-cylinder and a right prism. The base

ABDE of the container lies on a horizontal table. Right angled triangle GAB is the uniform cross-

section of the prism. The height of the container is 21 cm. The container is filled with water to a

height of 14cm and LM = 3 cm.

Calculate the volume, in cm3, of water in the container. [use π =

7

22] [4 marks]

Nov 2005, Q6

5. Diagram shows a solid cone with radius 9 cm and height 14 cm. A cylinder with radius 3 cm

and height 7 cm is taken out of the solid.

Calculate the volume, in cm3 , of the remaining solid.

[Use 7

22=π ] [4 marks]

Volume of Solids 17

July 2006

6. Diagram shows a solid cuboid. A cone is taken out of the solid. The diameter of the base of the

cone is 7 cm and the height of the cone is 9 cm. Calculate the volume of the remaining solid.

[Use 7

22=π ]. [4 marks ]

Nov 2006, Q5

7. Diagram 2 shows a combined solid

consists of a right prism and a right pyramid

which are joined at the plane EFGH. V is

vertically above the base EFGH. Trapezium

ABGF is the uniform cross of the prism.

The height of the pyramid is 8cm and FG = 14cm.

(a) Calculate the volume, in cm3, of the right pyramid

(b) It is given that the volume of the combined solid is 584cm3.

Calculate the length, in cm, of AF. [ 4 marks ]

10 cm

15 cm

12 cm

Volume of Solids 18

June 2007

Diagram 4 shows a solid right prism with a half-cylinder removed from the prism. The diameter of the

half-cylinder is 7 cm PQ = QR = 8 cm.

Calculate the volume, in cm3, of the solid. [Using

7

22 =π ] [ 4 marks ]

Nov 2007 , Q11 Diagram 6 shows a solid, formed by joining a cylinder to a right prism. Trapezium AFGB is the

uniform cross-section of the prism.

AB = BC = 9 cm. .The height of the cylinder is 6 cm and its diameter is 7 cm.

Calculate the volume, in cm3, of the solid. [Using

7

22 =π ] [ 4 marks ]

Diagram 6

E

C

B

D

G F

A

12 cm

8 cm

H

R

Q

P

12 cm

Diagram 4

Volume of Solids 19

June 2008 , Q8

Diagram 8 shows a composite solid. ABCDEFGH a right prism with trapezium ABGF as its cross-

section. AJBCKD is a half circular cylinder with diameter 14 cm. They joined at the rectangular plane ABCD.

Using 7

22 =π , calculate the volume, in cm

3, of the composite solid.

[ 4 marks ]

Nov 2008, Q4

Diagram 4 shows a composite solid formed by the combination of a right prism and half circular

cylinder at the rectangular plane ABFE. Right angled triangle DFE is the uniform cross-section of the

prism.

The diameter of the half circular cylinder is 7 cm and the volume of the composite solid is 451.5 cm3.

Using 7

22 =π , calculate

a) the volume , in cm3, of the half circular cylinder,

b) the length, in cm, of BC. [ 5 marks }

E

H

C

3 cm

B

D

G

F

A

J

8 cm

5 cm

Diagram 8

K

B

D

C

E

G

A H

6 cm

Diagram 4

F

Volume of Solids 20

ANSWERS

Chapter 2 Volume of Solids

Exercise 2.1

1 152 2 4 3 27

Exercise 2.2

1 4 2 3 3 36

Exercise 2.3 1 10 2 570 3 3

Exercise 2.4 1 251.43 2 84.86 3 7

Exercise 2.5 1 3 2 80 3 1.062

Exercise 2.6 1 6 2 14 3 2293.5

Exercise 2.7

1 4851 2 10.5 3 14

Exercise 2.8 1 718.67 2 21 3

21

2266

Exercise 2.9 Questions based on examination format

2 184.52 cm3 3 718.67 cm

3 4 792 cm

3 5 19 cm

3

6 569.71 cm3 7 1359.43 cm

3 8 576.19 cm

3 9 1448 cm

3

10 840 cm3

Exercise 2.10 Past Years SPM Questions

No Year Key No Year Key

1 2003N 402.5 11 2008N a)115.5

b) 16

2 2004J 91

3 2004N h = 6

4 2005J 1800

5 2005N 990

6 2006J 1710.17

7 2006N a)522.67

b) 2.37

8 2007J 153

9 2007N 987

10 2008J 1064