GCSE
Mathematics
Targeting Grade C
SSM 5
Volume
Can you…
If not you need
•Calculate the volume of a cuboid?
•Can you calculate the volume of prisms
(triangular, cylinder, trapezoidal)?
•Solve problems involving volumes of 3D
shapes?
•Can you work out a length if you have been
given the volume?
Practice 1: Remember the formula
for volume of a cuboid. You will be
given the formulae for the area of a
trapezium and the volume of a
prism, but you will need to know
the formulae for the area of a circle
and a triangle.
Practice 2: Decide which formulae
to use when calculating the
volume.
Practice 3: Substitute the values
given into the correct formula.
Try some questions
Questions:
Volume of a Cuboid
The formula for the volume of a cuboid is
Volume = length x breadth x height
V=lxbxh
Example
Note that there
are 3
dimensions so
the units are
cubic m or cm.
• Work out the volume
of this cuboid
10 cm
6 cm
15 cm
V=lxbxh
V = 15 x 6 x 10
V = 900cm³
• A cuboid has
• Height = 3m
• Length = 9m
• Breadth =5m
• What is its volume?
V=lxbxh
V=3x9x5
V = 135 m³
Practice 1
•
•
•
•
•
•
Maxine has two boxes in the
shape of cuboids.
Box A measures 12·3cm by
6cm by 3cm
Box B measures 9cm by 8·7cm
by 2·8cm
Maxine wants to use the box
with the greater volume
Give the letter of the box
Maxine should use
You must show all your
calculations
Answer
Practice 1 answer
Box A
V=lxbxh
V = 12.3 x 6 x 3
V = 221.4 cm³
Box B
V=lxbxh
V = 9 x 8.7 x 2.8
V = 219.24cm³
Box A has the greatest value
Area of Circle
The formula for the area of a circle
Area = pi x radius x radius
A=πxrxr
A = πr²
Area of a triangle
The formula for the area of a triangle is
Area = ½ x base x perpendicular height
A = ½ x b x h.
A = ½bh
Example
Calculate the volume of this trapezoidal
prism
Volume of prism is
Area of cross section x length
19.62 x 10
196.2cm³
10cm
10cm
Area of cross section (trapezium) is
½ (a+b)h
½ (4.2 + 6.7) x 3.6
½ x 10.9 x 3.6
½ x 39.24
19.62cm²
Remember
2 dimensions - units²
3 dimensions – units³
Example
Area of cross section (triangle)
is
½bxh
½x3x4
½ x 12 = 6cm²
Volume of prism is
Area of cross section x length
6 x 11 = 66cm³
Practice 2
(b)
(a)
The cylinder has a radius of 4cm and a
height of 15cm. Calculate the volume
of the cylinder. Give your answer
correct to 3 significant figures.
(Take π=3.14)
15cm
4cm
BC = 4cm, CF = 12cm, AB =
5cm and angle ABC is 90°.
Calculate the volume of the
triangular prism.
5cm
Answer
Practice 2 answer
(a)
Volume of cylinder = area of cross section x h
V = π r² h (Take π = 3.14)
V=πxrxrxh
V = 3.14 x 4 x 4 x 15
V = 753.6
V = 754 cm³ (3 SF)
(b)
Volume of prism = area of cross section x L
V=½xbxhxL
V = ½ x 4 x 5 x 12
V = 120 cm³
Practice 3A
Answer
(b) Calculate the volume of the silver bar.
Practice 3A answer
Area of cross section (trapezium)
½ (a + b) x h
½ (4 + 10) x 4
½ x 14 x 4
28cm²
Volume of silver bar
28 x 15
420cm³
Practice 3B
A box in the shape of a cube
has sides of length 2 cm.
8cm
10cm
2cm
6cm
These cube boxes are placed into
a larger cuboid box with dimensions
Answer
Height = 8cm
Length = 10cm
Width = 6cm
How many cube boxes fit into the cuboid box exactly?
Practice 3B answer
Volume of small cube
V=lxwxh
V=2x2x2
V = 8cm³
Volume of large cube
V = 6 x 8 x 10
V = 480 cm³
Number of small cubes in cuboid
480 ÷ 8 = 60
Example
The base of a cuboid is 10 cm by 10cm.
The volume of the cuboid is 1420cm³.
Find the height of the cuboid.
h cm
V =
1420 =
1420 =
1420 ÷ 100 =
14.2 =
lxbxh
10 x 10 x h
100 x h
h
h
So the height of the cuboid is 14.2cm.
10 cm
10 cm
Practice 4A
BC = 4cm, CF = 12 cm and
angle ABC = 90°.
If the volume of the triangular
prism is 84 cm³. What is the
length of the side AB of the
prism?
Practice 4A answer
Volume of prism = area of cross section x length
84 = ½ x 4 x AB x 12
84 = AB x 24
84 ÷ 24 = AB
AB = 3.5cm
Practice 4B
A cuboid has a volume of 160 cm³.
Its length is 8cm and its height is
4 cm.
Work out the breadth of the cuboid.
Answer
Practice 4B answer
Volume of cuboid = l x b x h
160 = 8 x 4 x b
160 = 32 x b
160 ÷ 32 = b
b = 5 cm
Questions
1. (a) Christopher buys a fish tank.
The dimensions of the tank are 91 cm
by 32 cm by 35 cm.
Answers
35cm
91cm
32cm
(i) Calculate the volume of the tank in
cm³.
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(ii) How many litres of water will the tank
hold when full?
(1000 cm³ = 1 litre)
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(2)
(i) Volume of a cuboid = l x b x h
V = 91 x 32 x 35
V = 101920 cm³
(ii) 101920 ÷ 1000 = 101.92 litres
The diagram shows a cuboid.
Answer
Volume of a cuboid = l x b x h
V=lxbxh
Height
Not to scale
8cm
180 = 8 x 5 x h
5cm
180 = 40 x h
The cuboid has a volume of 180 cm3.
Calculate the height of the cuboid.
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Answer .................................................... cm
180 ÷ 40 = h
4.5 = h
The height of the cuboid is 4.5cm
The diagram shows a bale of straw.
The bale is a cylinder with radius 70
cm and height 50 cm.
Answer
Volume of a cylinder
70 cm
= area of cross section x height
50 cm
Not to scale
Area of cross section = πr²
= π x 70 x 70
= π x 4900
Calculate the volume of the bale.
State your units.
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Answer ...........................
(4)
Volume = π x 4900 x h
= π x 4900 x 50
= π x 245000
= 769689.55
= 769690 cm³
= 769.69 m³
Answers
The diagram is a drawing of a triangular prism.
A
D
2 cm
B
(a)
(2)
(b)
(2)
5 cm
6 cm
C
Calculate the area of triangle ABC.
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Calculate the volume of the prism.
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Area of a triangle = ½ x base x height
=½x6x2
= ½ x 12
= 6 cm² (area of cross section)
Volume of prism = Area of cross section x length
V=6x5
V = 30 cm³
The diagram shows a
ridge tent which is 3.6m
long.
Calculate the volume of
the ridge tent.
Answer
Area of cross section
Area of rectangle = 1.9m x 0.8m
= 1.52m²
Area of triangle = ½ x 1.9m x 1.6m(2.4 –
0.8)
= ½ x 3.04
= 1.52m²
Area cross section = 1.52 + 1.52 = 3.04m²
2.4m
Volume of prism = Area of cross section x
0.8m
length
3.6m
= 3.04 x3.6
1.9m
= 10.944m³