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GCSE Mathematics. Targeting Grade C. SSM 5 Volume . If not you need. Can you… Calculate the volume of a cuboid? Can you calculate the volume of prisms (triangular, cylinder, trapezoidal)? Solve problems involving volumes of 3D shapes?

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Targeting Grade C


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slide1

GCSE Mathematics

Targeting Grade C

SSM 5

Volume

slide2

If not you need

  • Can you…
  • 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?
  • Try some questions

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.

Questions:

volume of a cuboid
Volume of a Cuboid

The formula for the volume of a cuboid is

Volume = length x breadth x height V = l x b x h

slide4
Work out the volume of this cuboid

V = l x b x h

V = 15 x 6 x 10

V = 900cm³

A cuboid has

Height = 3m

Length = 9m

Breadth =5m

What is its volume?

V = l x b x h

V = 3 x 9 x 5

V = 135 m³

10 cm

6 cm

15 cm

Note that there are 3 dimensions so the units are cubic m or cm.

Example

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

Practice 1

Answer

practice 1 answer
Practice 1 answer

Box A

V = l x b x h

V = 12.3 x 6 x 3

V = 221.4 cm³

Box B

V = l x b x h

V = 9 x 8.7 x 2.8

V = 219.24cm³

Box A has the greatest value

area of circle
Area of Circle

The formula for the area of a circle

Area = pi x radius x radius

A = π x r x r

A = πr²

area of a triangle
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

Example

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

example10
Example

Area of cross section (triangle) is

½ b x h

½ x 3 x 4

½ x 12 = 6cm²

Volume of prism is

Area of cross section x length

6 x 11 = 66cm³

practice 2
(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)

Practice 2

(b)

BC = 4cm, CF = 12cm, AB = 5cm and angle ABC is 90°.

Calculate the volume of the triangular prism.

15cm

5cm

4cm

Answer

practice 2 answer
Practice 2 answer

(a)

Volume of cylinder = area of cross section x h

V = π r² h (Take π = 3.14)

V = π x r x r x h

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 = ½ x b x h x L

V = ½ x 4 x 5 x 12

V = 120 cm³

practice 3a
Practice 3A

Answer

(b) Calculate the volume of the silver bar.

practice 3a answer
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

8cm

10cm

2cm

6cm

Practice 3B

A box in the shape of a cube

has sides of length 2 cm.

These cube boxes are placed into

a larger cuboid box with dimensions

Height = 8cm

Length = 10cm

Width = 6cm

How many cube boxes fit into the cuboid box exactly?

Answer

practice 3b answer
Practice 3B answer

Volume of small cube

V = l x w x h

V = 2 x 2 x 2

V = 8cm³

Volume of large cube

V = 6 x 8 x 10

V = 480 cm³

Number of small cubes in cuboid

480 ÷ 8 = 60

example17
The base of a cuboid is 10 cm by 10cm.

The volume of the cuboid is 1420cm³.

Find the height of the cuboid.

V = l x b x h

1420 = 10 x 10 x h

1420 = 100 x h

1420 ÷ 100 = h

14.2 = h

So the height of the cuboid is 14.2cm.

Example

h cm

10 cm

10 cm

practice 4a
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
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
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
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

slide23
1. (a) Christopher buys a fish tank.

The dimensions of the tank are 91 cm by 32 cm by 35 cm.

(i) Calculate the volume of the tank in cm³.

............................................................................................................................................................................................................................................

(ii) How many litres of water will the tank hold when full?

(1000 cm³ = 1 litre)

...................................................................................................................................................................................................................................................... (2)

Answers

  • Volume of a cuboid = l x b x h
  • V = 91 x 32 x 35
  • V = 101920 cm³

(ii) 101920 ÷ 1000 = 101.92 litres

slide24
The diagram shows a cuboid.

The cuboid has a volume of 180 cm3.

Calculate the height of the cuboid.

..............................................................................................................................................

................................................................................................................................................

................................................................................................................................................

................................................................................................................................................

Answer .................................................... cm

Answer

Volume of a cuboid = l x b x h

V = l x b x h

180 = 8 x 5 x h

180 = 40 x h

180 ÷ 40 = h

4.5 = h

The height of the cuboid is 4.5cm

slide25
The diagram shows a bale of straw.

The bale is a cylinder with radius 70

cm and height 50 cm.

Calculate the volume of the bale.

State your units.

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Answer ........................... (4)

Answer

Volume of a cylinder

= area of cross section x height

Area of cross section = πr²

= π x 70 x 70

= π x 4900

Volume = π x 4900 x h

= π x 4900 x 50

= π x 245000

= 769689.55

= 769690 cm³

= 769.69 m³

slide26
The diagram is a drawing of a triangular prism.

(a)Calculate the area of triangle ABC.

.....................................................................................................................................................................................................................................................................................................................................................................................................

(2)

(b)Calculate the volume of the prism.

.....................................................................................................................................................................................................................................................................................................................................................................................................

(2)

Answers

Area of a triangle = ½ x base x height

= ½ x 6 x 2

= ½ x 12

= 6 cm² (area of cross section)

Volume of prism = Area of cross section x length

V = 6 x 5

V = 30 cm³

slide27
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

Volume of prism = Area of cross section x

length

= 3.04 x3.6

= 10.944m³

2.4m

0.8m

3.6m

1.9m