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3D trigonometry Area Area of a circle Substitution Rounding

This topic is relevant for:

GCSE Maths Geometry and Measure 3D Shapes

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 3D 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

                Cylinder

             How to calculate volume Image3

                Pyramid



                Cone



                Sphere

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

           $(\text{Volume }=$
             $\text{Area of rectangle } \times \text{ Length})$

           $\text{Volume }=$
             $\text{Area of cross section} \times \text{ Length}$

           $\text{Volume }=\pi{r}^2{h}$

           $(\text{Volume }=$
             $\text{Area of circle }\times\text{ Height})$

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

           $\text{Volume }=\frac{1}{3}\pi{r}^2{h}$

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

             $\text{Volume }=\frac{4}{3}\pi{r}^3$

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.

Step-by-step guide: Volume formula

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:

Write down the formula.
Substitute the values into the formula.
Complete the calculation.
Write the answer, including the units.

Explain how to calculate volume

Volume worksheet

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

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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:

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 $(cm^3)$ .

$V$ = $135cm^3$

Step-by-step guide: Volume of a cuboid

Example 2: volume of a prism

Calculate the volume of this triangular prism:

Write down the formula.

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

Substitute the values into the formula.

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:

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

Complete the calculation.

$V$ = $42$

Write the answer, including the units.

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

$V=42m^3$

Step-by-step guide: Volume of a prism

Example 3: volume of a cylinder

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

Write down the formula.

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

Substitute the values into the formula.

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

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

Complete the calculation.

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

Write the answer, including the units.

$V$ = $549.78cm^3$

Step-by-step guide: Volume of a cylinder

Example 4: volume of a pyramid

Work out the volume of the pyramid:

Write down the formula.

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

Substitute the values into the formula.

First we need to find the area of the base

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

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

Complete the calculation.

$V$ = $84$

Write the answer, including the units.

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

$V=84cm^3$

Step-by-step guide: Volume of a pyramid

Example 5: volume of a cone

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

Write down the formula.

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

Substitute the values into the formula.

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

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

Complete the calculation.

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

Write the answer, including the units.

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

$V=56.5mm^3$

Step-by-step guide: Volume of a cone

Example 6: volume of a sphere

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

Write down the formula.

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

Substitute the values into the formula.

The radius of the sphere is $9cm.$

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

Complete the calculation.

$V$ = $3053.628059...$

Write the answer, including the units.

$V$ = $3054cm^3$

Step-by-step guide: Volume of a sphere

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

$70cm^3$

$166cm^3 $

$140cm^3 $

$35cm^3 $

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

$300cm^3$

$420cm^3 $

$320cm^3 $

$210cm^3 $

\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}

$125m^3$

$314m^3 $

$754m^3 $

$628m^3 $

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

$36cm^3$

$12cm^3 $

$18cm^3 $

$4cm^3 $

\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}

$366.5mm^3$

$1099.6mm^3 $

$146.6mm^3 $

$70.0mm^3 $

\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}

$151cm^3$

$226cm^3 $

$113cm^3 $

$905cm^3 $

\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

(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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Introduction

What is volume?

How to calculate volume

Volume worksheets

Volume examples

↓

Example 1: volume of a cuboid Example 2: volume of a prism Example 3: volume of a cylinder Example 4: volume of a pyramid Example 5: volume of a cone Example 6: volume of a sphere

Common misconceptions

Practice volume questions

Volume GCSE questions

Learning checklist

Next lessons

Sill stuck?