# 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. he pyramid could have a square base, a triangular base, it could be a pentagonal pyramid or a hexagonal pyramid. The volume formula is the same no matter what the shape of the base.

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.

This formula can be applied to any pyramid where the base is a polygon. It can also be used to find the volume of a cone.

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}

Note:

The volume of any pyramid is one third of the volume of a prism with the same base shape and height h.

### 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 ## Volume of a pyramid examples

### Example 1: calculating the volume with a diagram included

Calculate the volume of the pyramid below.

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.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### 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 .

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### 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.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### 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.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Example 6: calculating the height

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

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Common misconceptions

• The height is the perpendicular height

The height of the pyramid needs to be the perpendicular height not the slant height.  This is the height that is at a right-angle to the base. You may need to find this height by using trigonometry or Pythagoras’ Theorem.

• Volume has cubic units

The volume will have cube units such as cubic centimetres ( cm^3 ) or cubic metres ( 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. 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.\dot{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}

### 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)

\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.

(1 mark)

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)

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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