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Faces, edges and vertices

SubstitutionArea of triangles

Area of quadrilaterals Volume of prismsThis topic is relevant for:

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.

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.

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

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.

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.

\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}In order to calculate the volume of a pyramid:

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

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

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

COMING SOONCalculate the volume of the pyramid below.

**Calculate the area of the base.**

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

\text{Area of base}=7\times 7=492**Substitute 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}3**Write the answer, including the units.**

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

**Calculate the area of the base.**

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

**Substitute values into the formula and solve.**

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}

**Write the answer, including the units.**

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

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

**Calculate the area of the base.**

The base is a square with side 8 \ m .

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

**Substitute values into the formula and solve.**

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}

**Write the answer, including the units.**

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

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.

**Calculate the area of the base.**

The base is a square with side 12.3 \ mm .

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

**Substitute values into the formula and solve.**

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}

**Write the answer, including the units.**

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

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.

**Calculate the area of the base.**

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

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

**Substitute values into the formula and solve.**

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}

**Write the answer, including the units.**

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

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

**Calculate the area of the base.**

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

**Substitute values into the formula and solve.**

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}

**Write the answer, including the units.**

h=10\text{ cm}

**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:

1. Calculate the volume of the pyramid below.

45 \text{ cm}^3

15 \text{ cm}^3

11 \text{ cm}^3

30 \text{ cm}^3

\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:

229.5 \text{ mm}^3

668.8 \text{ mm}^3

229.6 \text{ mm}^3

229.6 \text{ cm}^3

\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

220 \text{ m}^3

360 \text{ m}^3

120 \text{ m}^3

\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

33.6 \text{ cm}^3

29.4 \text{ cm}^3

444.1 \text{ cm}^3

\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:

35 \text{ cm}^3

25 \text{ cm}^3

105 \text{ cm}^3

49 \text{ cm}^3

\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}

7 \text{ cm}

2.7 \text{ cm} \ (1dp)

0.\overline{8}\text{ cm}

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}

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.

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)**

You have now learned how to:

- Calculate volumes of pyramids

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