Volume of a Pyramid

Here we will learn about the volume of a pyramid, including how to find the volume of a pyramid using the base.

There are also volume of a pyramid 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 the volume of a pyramid?

The volume of a pyramid is how much space there is inside a pyramid.

For example, the great pyramids of Giza have an approximate volume of 2600000m^3 .

To calculate the volume of a pyramid, we need to know certain lengths of the pyramid.

A pyramid is a three dimensional shape made up of flat faces.  It has a base and triangular faces which meet at a point, called the apex.  The vertical height is the length from the base to the apex and is perpendicular to the base of the pyramid.

Volume of a pyramid Image 1

To calculate the volume of a pyramid, we use the formula:

V=\frac{1}{3}Bh

where:

  • V represents the volume of the pyramid,
  • B represents the area of the base of the pyramid,
  • h represents the perpendicular height of the pyramid.

Volume of a pyramid Image 2

This formula can be applied to any pyramid where the base is a polygon.

E.g. calculate the volume of a square-based pyramid where the base has a side length of 4 \ cm and the height of the pyramid is 6 \ cm.

Volume of a pyramid Image 3

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times(4\times{4})\times{6}\\\\ &=32\text{ cm}^{3} \end{aligned}

What is the volume of a pyramid?

What is the volume of a pyramid?

How to calculate the volume of a pyramid

In order to calculate the volume of a pyramid:

  1. Calculate the area of the base.
  2. Substitute values into the formula and solve.
  3. Write the answer, including the units.

How to calculate the volume of a pyramid

How to calculate the volume of a pyramid

Volume of a pyramid worksheet

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

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Volume of a pyramid worksheet

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

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Volume of a pyramid examples

Example 1: calculating the volume with a diagram included

Calculate the volume of the pyramid below.

Volume of a pyramid Example 1

  1. Calculate the area of the base.

The base is a square with side length 7 \ cm .

\text{Area of base}=7\times 7=49

2Substitute values into the formula and solve.

As B=49cm^2 and h=9cm , substituting these into the formula for the volume of a pyramid, we get:

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{49}\times{9}\\\\ &=147 \end{aligned}

3Write the answer, including the units.

V=147\text{ cm}^{3}

Example 2: rectangular based pyramid

Calculate the volume of this pyramid. Write your answer to 1 decimal place.

Volume of a pyramid Example 2

The base is a rectangle with side lengths 6.2 \ cm and 5.3 \ cm.


\text{Area of base}=6.2\times{5.3}=32.86

As B=32.86cm^2 and h=8.5cm , substituting these into the formula for the volume of a pyramid, we get:


\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{32.86}\times{8.5}\\\\ &=93.10333333... \end{aligned}

V=93.1\text{ cm}^{3}\text{ (1dp)}

Example 3: without a diagram

Calculate the volume of a square-based pyramid where the side length of the base is 8 \ m and perpendicular height is 12 \ m .

The base is a square with side 8 \ m .


\text{Area of base}=8\times 8=64

As B=64m^2 and h=12m , substituting these into the formula for the volume of a pyramid, we get:


\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{64}\times{12}\\\\ &=256 \end{aligned}

V=256\text{ cm}^{3}

Example 4: without a diagram

Find the volume of a square-based pyramid with the side length of the base equal to 12.3 \ mm , and the height of the pyramid equal to 18.2 \ mm .  Give your answer correct to 2 decimal places.

The base is a square with side 12.3 \ mm .


\text{Area of base}=12.3\times{12.3} =151.29

As B=151.29mm^2 and h=18.2mm , substituting these into the formula for the volume of a pyramid, we get:


\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{151.29}\times{18.2}\\\\ &=917.826 \end{aligned}

V=917.83\text{ mm}^{3}\text{ (2dp)}

Example 5: rectangular base

ABCDE is a rectangular base pyramid. The point F is the centre of the base, directly below the vertex E , and G is the midpoint of the line AD . Given that EF=6cm and DG=2.5cm, calculate the volume of this pyramid.

Volume of a pyramid Example 5

The base is a rectangle where AD=2 \times DG=5cm and CD=4cm .


\text{Area of base}=5\times{4} =20

As B=20cm^2 and h=6cm , substituting these into the formula for the volume of a pyramid, we get:


\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{20}\times{6}\\\\ &=40 \end{aligned}

V=40\text{ cm}^{3}

Example 6: calculating the height

Calculate the height of a pyramid with volume 40 \ cm^3 and base area 12 \ cm^3 .

Here, we already know the base has an area of 12cm^2 so we can move on to step 2 .

As B=12cm^2 and V=40cm^3 , substituting these into the formula for the volume of a pyramid, we get:


\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ 40&=\frac{1}{3}\times{12}\times{h}\\\\ 40&=4h\\\\ &h=10 \end{aligned}

h=10\text{ cm}

Common misconceptions

  • The height is the perpendicular height

The height of the pyramid needs to be the perpendicular height.  This is the height that is at a right-angle to the base.

  • Volume has cube units

The volume will have cube units such as cubic centimetres ( cm^3 ) or cubic meters ( m^3 ).

  • Be accurate

When there are two or more steps in your workings, do not round your workings.  For example – do not round the area of the base.  Only round at the end of the question so that your answer is accurate.

  • Take care with rounding

At the end of the question, make sure you round your answer to the correct number of decimal places or significant figures.

Volume of a pyramid is part of our series of lessons to support revision on pyramids. You may find it helpful to start with the main pyramid lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice volume of a pyramid questions

1. Calculate the volume of the pyramid below.

 

Volume of a pyramid Practice Question 1

45 \text{ cm}^3
GCSE Quiz False

15 \text{ cm}^3
GCSE Quiz True

11 \text{ cm}^3
GCSE Quiz False

30 \text{ cm}^3
GCSE Quiz False
\text{Area of base}=3\times{3} =9

 

As B=9cm^2 and h=5cm ,

 

\begin{aligned} V&=\frac{1}{3}\times{b}\times{h} \\\\ &=\frac{1}{3}\times{9}\times{5} \\\\ &=15 \end{aligned}

 

V=15\text{ cm}^{3}

2. Find the volume. Give your answer correct to 1 decimal place:
Volume of a pyramid Practice Question 2

229.5 \text{ mm}^3
GCSE Quiz False

668.8 \text{ mm}^3
GCSE Quiz False

229.6 \text{ mm}^3
GCSE Quiz True

229.6 \text{ cm}^3
GCSE Quiz False
\text{Area of base}=8.7\times{8.7} =75.69

 

As B=75.69mm^2 and h=9.1mm ,

 

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{75.69}\times{9.1}\\\\ &=229.593 \end{aligned}

 

V=229.6\text{ mm}^{3}

3. Find the volume of a square-based pyramid where the side length of the base is 6 \ m and the height is 10 \ m.

300 \text{ m}^3
GCSE Quiz False

220 \text{ m}^3
GCSE Quiz False

360 \text{ m}^3
GCSE Quiz False

120 \text{ m}^3
GCSE Quiz True
\text{Area of base}=6\times{6} =36

 

As B=36m^2 and h=10m ,

 

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{36}\times{10}\\\\ &=120 \end{aligned}

 

V=120\text{ m}^{3}

4. ABCDE is a square based pyramid. ABCD is the base of the pyramid. E is the apex, directly above the centre of the base.

 

Given that AB=3.5cm and h=7.2cm, calculate the volume of the pyramid.

16.3 \text{ cm}^3
GCSE Quiz False

33.6 \text{ cm}^3
GCSE Quiz False

29.4 \text{ cm}^3
GCSE Quiz True

444.1 \text{ cm}^3
GCSE Quiz False
\text{Area of base}=3.5\times{3.5} =12.25

 

As B=12.25cm^2 and h=7.2cm ,

 

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{12.25}\times{7.2}\\\\ &=29.4 \end{aligned}

 

V=29.4\text{ cm}^{3}

5. Calculate the volume of this pyramid:

 

Volume of a pyramid Practice Question 5

35 \text{ cm}^3
GCSE Quiz True

25 \text{ cm}^3
GCSE Quiz False

105 \text{ cm}^3
GCSE Quiz False

49 \text{ cm}^3
GCSE Quiz False
\text{Area of base}=7\times{5} =35

 

As B=35cm^2 and h=3cm ,

 

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ &=\frac{1}{3}\times{35}\times{3}\\\\ &=35 \end{aligned}

 

V=35\text{ cm}^{3}

6. Find the height of a pyramid with volume 56 \ cm^3 and base area 21 \ cm^2.

8 \text{ cm}
GCSE Quiz True

7 \text{ cm}
GCSE Quiz False

2.7 \text{ cm} \ (1dp)
GCSE Quiz False

0.\overline{8}\text{ cm}
GCSE Quiz False

As V=56cm^3 and B=21cm^2,

 

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ 56&=\frac{1}{3}\times{21}\times{h}\\\\ 56&=7h\\\\ &h=8 \end{aligned}

 

h=8\text{ cm}

Volume of a pyramid GCSE questions

1. A square-based pyramid has a base with side length 12 \ cm.

The perpendicular height of the pyramid is 17 \ cm.

 

Calculate the volume of the pyramid. State the units of your answer.

 

(3 marks)

Show answer
\frac{1}{3}\times 12^2 \times 17

(1)

 

816

(1)

 

\text{cm}^{3}

(1)

2. A pyramid has the volume 460 \ cm^3 and a base with the area 200 \ cm^2.

Calculate the height of the pyramid.

 

Circle the correct answer:

 

\begin{aligned} &A \quad \quad \quad \quad \quad B \quad \quad \quad \quad \;\; C \quad \quad \quad \quad \;\; D \\ 6.&5 \ cm \quad \quad \;\; 6.7 \ cm \quad \quad \;\; 6.9 \ cm \quad \quad \;\; 7.1 \ cm \end{aligned}

(1 mark)

Show answer
C \ – \ 6.9 \ cm

(1)

3. A solid is made from a pyramid on top of a cuboid.

 

Volume of a pyramid GCSE Question 3

 

Calculate the volume of the compound 3D shape.

 

(3 marks)

Show answer
6\times 5\times 4=120

(1)

 

\frac{1}{3}\times (6\times 5) \times 3=30

(1)

 

120+30=150

(1)

Learning checklist

You have now learned how to:

  • Calculate volumes of pyramids

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