GCSE Maths Geometry and Measure

Volume

How to Calculate Volume

How to Calculate Volume

Here we will learn how to calculate the volume of three-dimensional shapes, including cubes, cuboids, prisms, cylinders, pyramids, cones and spheres.

There are also volume worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is volume?

The volume of a 3 D shape is the amount of space there is inside the shape.

To calculate the volume of a shape in three dimensions, we can use the relevant volume formula.

The table below shows the formulae that we can use for some of the most common 3D shapes:

 
                Cuboid
 
  How to calculate volume Image 1
 
                Prism
 
How to calculate volume Image 2
 
                Cylinder
 
            How to calculate volume Image3
 
                Pyramid
 
        How to calculate volume Image 4
 
                Cone
 
            How to calculate volume Image 5
 
                Sphere
 
            

 
 
 
 
           \text{Volume }={h}\times{w}\times{d}
 
           (\text{Volume }=
             \text{Area of rectangle } \times \text{ Depth})
 
 
 
 
 
           \text{Volume }=
             \text{Area of cross section} \times \text{ Depth}
 
 
 
 
 
           \text{Volume }=\pi{r}^2{h}
 
           (\text{Volume }=
             \text{Area of circle }\times\text{ Height})
 
 
 
 
           \text{Volume }=
             \frac{1}{3}\times\text{Area of base }\times\text{Height}
 
 
 
 
 
 
           \text{Volume }=\frac{1}{3}\pi{r}^2{h}
 
 
           \text{Volume }=
             \frac{1}{3}\times\text{Area of base }\times\text{Height}
 
 
 
 
 
 
 
 
 
             \text{Volume }=\frac{4}{3}\pi{r}^3
 
 
 
 

The formulae for the volume of a cone and the volume of a sphere are given to you for GCSE maths. You need to learn the other formulae.

Volume is measured in cubic units, including metric units such as cm^3 (cubic centimetres), m^3 (cubic metres) or mm^3 (cubic millimetres). Volume could also be measured in imperial units, including cubic feet or cubic inches. It can also be described using units of capacity such as millilitres, litres, pints or gallons.

What is volume?

What is volume?

Volume of composite solid

You may be asked to calculate the volume of a 3 dimensional shape which is composed of 2 or more separate shapes. This type of shape is known as a composite shape. The volume of each of the shapes can be found and then added together.

For example,

Here is a composite shape. It is made of a cuboid and a pyramid.

The volume of the cuboid is

8\times 8\times 10=640

The volume of the pyramid is

\frac{1}{3} \times 8^2 \times 6=128

The total volume can be found by adding the volume of the cuboid and the volume of the pyramid.

640+128=768 \ cm^3

How to calculate volume

In order to calculate volume:

  1. Write down the formula.
  2. Substitute the values into the formula.
  3. Complete the calculation.
  4. Write the answer, including the units.

Explain how to calculate volume

Explain how to calculate volume

Volume worksheet

Volume worksheet

Volume worksheet

Get your free volume worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Volume worksheet

Volume worksheet

Volume worksheet

Get your free volume worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Volume examples

Example 1: volume of a cuboid

Calculate the volume of the cuboid below:

How to calculate volume Example 1

  1. Write down the formula.

The volume (V) of a cuboid is the same as the volume of a rectangular prism or the volume of a box.

\text{Volume } = {h}\times{w}\times{d}

2Substitute the values into the formula.

Given that h=3cm, \; w= 9cm, and d= 5cm, we have:

V = 3 \times 9 \times 5

3Complete the calculation.

V = 135

4Write the answer, including the units.

The dimensions of the cuboid are given in centimetres, so the volume will be in cubic centimetres (cm^3) .

V = 135cm^3

Example 2: volume of a prism

Calculate the volume of this triangular prism:

How to calculate volume Example 2

\text{Volume of prism } = \text{Area of cross section} \times \text{Depth}

First, we need to find the area of the cross section, which is a triangle:


\begin{aligned} \text{Area of a triangle}&=\frac{1}{2}\times b \times h\\\\ &=\frac{1}{2}\times 2 \times 7\\\\ &=7\text{ m}^2 \end{aligned}


The depth of the prism is 6m, therefore:


\begin{aligned} \text{Volume of prism }&= \text{Area of cross section} \times \text{ Depth}\\\\ &=7\times 6 \end{aligned}

V = 42

The dimensions are given in metres, so the volume is given in cubic metres (m^3):


V=42m^3

Example 3: volume of a cylinder

Find the volume of the cylinder. Give your answer to 2 decimal places.

How to calculate volume Example 3

\text{Volume of a cylinder} = \pi r^{2}h

The height of the cylinder is 7cm and the radius is 5cm.


V=\pi \times 5^{2} \times 7

V=\pi \times 5^{2} \times 7 = 549.7787144...

V = 549.78cm^3

Example 4: volume of a pyramid

Work out the volume of the pyramid:

How to calculate volume Example 4

\text{Volume of a pyramid} = \frac{1}{3} \times \text{ Area of base } \times \text{ Height}

First we need to find the area of the base


\begin{aligned} \text{Area of base }&=6 \times 6\\\\ &=36\text{cm}^2 \end{aligned}


\begin{aligned} \text{Volume of pyramid }&=\frac{1}{3} \times \text{ Area of base } \times \text{ Height}\\\\ &=\frac{1}{3} \times 36 \times 7 \end{aligned}

V = 84

The length units are centimetres so the units for the volume are cubic centimetres (cm^3).


V=84cm^3

Example 5: volume of a cone

Calculate the volume of the cone. Write your answer to 3 significant figures.

How to calculate volume Example 5

\text{Volume of a cone } = \frac{1}{3} \pi r^{2}h

The height of the cone is 6mm and the radius is 3mm.


V = \frac{1}{3} \times \pi \times 3^{2} \times 6

V = \frac{1}{3} \times \pi \times 3^{2} \times 6 = 56.54866776...

The dimensions are given in metres, so the volume is given in cubic millimetres (mm^3).


V=56.5mm^3

Example 6: volume of a sphere

Work out the volume of the sphere. Give your answer to the nearest integer.

How to calculate volume Example 6

\text{Volume of a sphere } = \frac{4}{3} \pi r^{3}

The radius of the sphere is 9cm.


V=\frac{4}{3} \times \pi \times 9^{3}

V = 3053.628059...

V = 3054cm^3

Common misconceptions

  • Missing/incorrect units

You should always include units in your answer. 

Volume is measured in cubic units (e.g. mm^3, cm^3, m^3 etc)

  • Calculating with different units

You need to make sure all measurements are in the same units before calculating the volume. (For example, you can’t have some measurements in cm and some in m ).

  • Make sure you get the correct formula

Be careful to make sure you use the correct formula for the volume of the 3 D shape.

  • Don’t round off too early

It is important to not round decimals until the end of the calculation. Rounding too early will result in an inaccurate answer.

Practice volume questions

1. Work out the volume of the cuboid

 

How to calculate volume Practice Question 1

 

70cm^3

GCSE Quiz False

166cm^3 ​

GCSE Quiz False

140cm^3 ​

GCSE Quiz True

35cm^3 ​

GCSE Quiz False
\begin{aligned} V=h\times{w}\times{d} &=4 \times 7 \times 5\\ &=140\mathrm{cm}^{3} \end{aligned}

2. Calculate the volume of the prism.

 

How to calculate volume Practice Question 2

 

300cm^3

GCSE Quiz True

420cm^3 ​

GCSE Quiz False

320cm^3 ​

GCSE Quiz False

210cm^3 ​

GCSE Quiz False
\text{Volume of a prism }=\text{Area of cross section } \times { Depth} \begin{aligned} \text{Area of trapezium }&=\frac{1}{2}(a+b)h\\\\ &=\frac{1}{2}\times (3+7)\times 6\\\\ &=30\text{cm}^2 \end{aligned} \begin{aligned} \text{Volume }&=30 \times 10\\\\ &=300\mathrm{cm}^{3} \end{aligned}

3. Calculate the volume of the cylinder. Give your answer to 3 significant figures.

 

How to calculate volume Practice Question 3

 

125m^3

GCSE Quiz False

314m^3 ​

 

GCSE Quiz False

754m^3 ​

GCSE Quiz False

628m^3 ​

GCSE Quiz True
\begin{aligned} \text{Volume }&=\pi r^{2}h\\\\ &=\pi \times 10^{2} \times 2\\\\ &=628.3185307…\\\\ &=628\mathrm{m}^{3} \end{aligned}

4. Find the volume of the pyramid  

 

How to calculate volume Practice Question 4

 

36cm^3

GCSE Quiz False

12cm^3 ​

GCSE Quiz True

18cm^3 ​

GCSE Quiz False

4cm^3 ​

GCSE Quiz False
\text{Volume of a pyramid }=\frac{1}{3} \times \text{ Area of base } \times { Height } \begin{aligned} \text{Area of base }&=3 \times 3\\\\ &=9\text{cm}^2 \end{aligned} \begin{aligned} \text{Volume }&=\frac{1}{3} \times 9 \times 4\\\\ &=12\mathrm{cm}^{3} \end{aligned}

5. Find the volume of the cone. Give your answer to 1 d.p.

 

How to calculate volume Practice Question 5

 

366.5mm^3

GCSE Quiz True

1099.6mm^3 ​

GCSE Quiz False

146.6mm^3 ​

GCSE Quiz False

70.0mm^3 ​

GCSE Quiz False
\begin{aligned} \text{Volume }&= \frac{1}{3} \pi r^{2}h\\\\ &=\frac{1}{3} \times \pi \times 5^{2} \times 14\\\\ &=366.5191429…\\\\ &=366.5\mathrm{mm}^{3} \end{aligned}

6.  Calculate the volume of the sphere. Give your answer to  3 s.f.

 

How to calculate volume Practice Question 6

 

151cm^3

GCSE Quiz False

226cm^3 ​

GCSE Quiz False

113cm^3 ​

GCSE Quiz False

905cm^3 ​

GCSE Quiz True
\begin{aligned} \text{Volume }&=\frac{4}{3}\pi r^{3}\\\\ &=\frac{4}{3} \times \pi \times 6^{3}\\\\ &=904.8\mathrm{cm}^{3} \end{aligned}

Volume GCSE questions

1. Calculate the volume of this cylinder. Give your answer to 1 decimal place.

 

How to calculate volume GCSE Question 1

 

(3 marks)

Show answer

 
Radius = 5.5cm

(1)

 
 
\begin{aligned} V&=\pi r^{2}h \\\\ &=\pi \times 5.5^{2} \times 7 \end{aligned}

(1)

 
 
\begin{aligned} V&=665.2322444… \\\\ &=665.2\text{cm}^3 \end{aligned}

(1)

 

2. (a) Calculate the volume of this container

 

How to calculate volume GCSE Question 2

 

(b) Three spheres, each with radius 0.6m, are placed inside the container. What percentage of the space in the container do the spheres take up?

 

(5 marks)

Show answer

(a)
 
\text{Area of cross-section: } \frac{1}{2}(1.4+2)\times 1.2=2.04

(1)

 
 
\text{Volume of prism: }2.04 \times 4 = 8.16\mathrm{m}^{3}

(1)

 
 
(b)
 
 
\text{Volume of one sphere: }\frac{4}{3} \times \pi \times 0.6^{3}=0.9048\mathrm{m}^{3}

(1)

 
 
\text{Total volume of three spheres: } 3 \times 0.9048=2.7144\mathrm{m}^{3}

(1)

 
 
\text{Percentage of space: }\frac{2.7144}{8.16} \times 100=33.3\%

(1)

 

3. Both of these cuboids have the same volume. Work out the height of cuboid B .

 

How to calculate volume GCSE Question 3 Image 1 How to calculate volume GCSE Question 3 Image 2

 

(4 marks)

Show answer

 
\text{Volume of A: } 14 \times 6 \times 5=420\mathrm{cm}^{3}

(1)

 
 
\text{Volume of B: } 15 \times 7 \times h

(1)

 
 
\begin{aligned} 15 \times 7 \times{h}&=420\\\\ 105h&=420 \end{aligned}

(1)

 
 
h=4cm

(1)

 

Learning checklist

You have now learned how to:

  • Calculate the volume of cuboids and prisms
  • Calculate the volume of cylinders, pyramids, cones and spheres
  • Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3D

The next lessons are

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