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Volume Of Square Based Pyramid

Here we will learn about the volume of a square based pyramid, including how to calculate the volume of a square based pyramid and how to solve problems involving the volume.

There are also square based 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 volume of a square based pyramid?

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

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

Volume of square based pyramid image 1

To calculate the volume of a square based pyramid we use the formula

\text{Volume}=\cfrac{1}{3}\times \text{area of base} \times \text{height} .

The pyramid height should be perpendicular to its base.

The formula can also be written as

V=\cfrac{1}{3}Bh,

where

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

If we are not given the perpendicular height, we can use Pythagoras’ theorem to find it from the given slant height or the lateral edge length.

Volume of square based pyramid image 2

There are lots of different types of pyramids because the base of a pyramid can be any polygon.

For example, a rectangular pyramid has a rectangular base and a triangular pyramid (often known as a tetrahedron) has a triangular base.

The square pyramid formula for volume can be used for any pyramid.

What is volume of a square based pyramid?

What is volume of a square based pyramid?

How to calculate the volume of a square based pyramid

In order to calculate the volume of a square based pyramid:

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

Explain how to calculate the volume of a square based pyramid

Explain how to calculate the volume of a square based pyramid

Volume and surface area of a pyramid worksheet (includes volume of square based pyramid)

Volume and surface area of a pyramid worksheet (includes volume of square based pyramid)

Volume and surface area of a pyramid worksheet (includes volume of square based pyramid)

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

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Volume and surface area of a pyramid worksheet (includes volume of square based pyramid)

Volume and surface area of a pyramid worksheet (includes volume of square based pyramid)

Volume and surface area of a pyramid worksheet (includes volume of square based pyramid)

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

DOWNLOAD FREE

Volume of square based pyramid examples

Example 1: calculating the volume, given the height, of a square pyramid

Find the volume of the square based pyramid.

Volume of square based pyramid example 1

  1. Calculate the area of the base.

The area of the base can be found by squaring the base edge.

9^2=81

2Substitute values into the formula and solve.

The volume of a square based pyramid can be found by using the formula.

\begin{aligned} V&=\cfrac{1}{3}Bh \\\\ V&=\cfrac{1}{3}\times 81\times 13 \\\\ V&=351 \end{aligned}

3Write the answer, including the units.

The volume is 351 \ m^{3}.

Example 2: calculating the volume, given the height, of a square pyramid

Find the volume of the square based pyramid.

Volume of square based pyramid example 2

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

Example 3: calculating the volume of a square pyramid not given perpendicular height

Find the volume of the square based pyramid. The apex of the pyramid is directly above the centre of the square base.

Volume of square based pyramid example 3 image 1

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

Example 4: calculating the volume of a square pyramid not given perpendicular height

Find the volume of the square based pyramid. The apex of the pyramid is directly above the centre of the square base.

Volume of square based pyramid example 4 image 1

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

Example 5: calculating the height given the volume

The volume of the square based pyramid is 87.6 \ cm^{3}. Find the perpendicular height, h. Give your answer to 3 significant figures.

Volume of square based pyramid example 5

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

Example 6: calculating the length of the base given the volume

The volume of the square based pyramid is 1200 \ m^{3}. Find the base edge. Give your answer correct to 1 decimal place.

Volume of square based pyramid example 6

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

  • Use the correct units

Remember to use cubic units for volume such as cm^{3} or m^{3}. Square units such as cm^{2} or m^{2} are for areas.

  • Use the correct volume formula

There are many different volume formulas in maths. Make sure that you use the correct one to work out the volume of a pyramid.

\text{Volume}=\cfrac{1}{3}\times \text{area of base} \times \text{height}

Or

V=\cfrac{1}{3}Bh

  • Be accurate

When there are two or more steps in your workings, do not round your workings. Only round at the end of your solution 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 square based pyramid questions

1. Work out the volume of this square based pyramid.

 

Volume of square based pyramid practice question 1

2484 \ m^3
GCSE Quiz True

2484 \ m^2
GCSE Quiz False

7452 \ m^3
GCSE Quiz False

7452 \ m^2
GCSE Quiz False

We can work out the volume of a pyramid by using the formula

 

V=\cfrac{1}{3}Bh.

 

The base area can be found by squaring the side length of the base square.

 

18^2=324

 

We can substitute the values we have been given into the formula and work out the volume.

 

V=\cfrac{1}{3}\times 324 \times 23=2484

 

So the volume is 2484 \ m^{3}.

2. Work out the volume of this square based pyramid. Give your answer in cm^{3}.

 

Volume of square based pyramid practice question 2

4522.5 \ cm^3
GCSE Quiz True

452 \ cm^3
GCSE Quiz False

45.2 \ cm^3
GCSE Quiz False

45 \ 225 \ mm^3
GCSE Quiz False

We can work out the volume of a pyramid by using the formula

 

V=\cfrac{1}{3}Bh.

 

The base area can be found by squaring the side length. The side length is in mm so needs converting the cm.

 

4.5^2=20.25

 

The height of 6.7 \ m needs to be converted to 670 \ cm .

 

We can substitute the values we have been given into the formula and work out the volume.

 

V=\cfrac{1}{3}\times 20.25 \times 670=4522.5

 

So the volume is 4522.5 \ cm^{3} .

3. Work out the volume of this square based pyramid. Give your answer in cm^{3} correct to three significant figures.

 

Volume of square based pyramid practice question 3

22 \ 200 \ cm^3
GCSE Quiz False

33 \ 300 \ cm^3
GCSE Quiz False

11 \ 100 \ cm^3
GCSE Quiz True

44 \ 400 \ cm^3
GCSE Quiz False

We can work out the volume of a pyramid by using the formula

 

V=\cfrac{1}{3}Bh.

 

The base area can be found by squaring the side length.

 

30^2=900

 

We have been given the slant length. We can use this to find the perpendicular height of the pyramid.

 

h=\sqrt{40^2-15^2}=5\sqrt{55}

 

We can substitute the values we have been given into the formula and work out the volume.

 

V=\cfrac{1}{3}\times 900 \times 5\sqrt{55}=11124.29…

 

So the volume is 11 \ 100 \ cm^{3} (to 3 sf).

4. Work out the volume of this square based pyramid. V is directly above B and VC is 10 \ cm. Give your answer in cm^{3}.

 

Volume of square based pyramid practice question 4

640 \ cm^3
GCSE Quiz False

128 \ cm^3
GCSE Quiz True

213 \ cm^3
GCSE Quiz False

384 \ cm^3
GCSE Quiz False

We can work out the volume of a pyramid by using the formula

 

V=\cfrac{1}{3}Bh.

 

The base area can be found by squaring the side length.

 

8^2=64

 

We have been given a slant length. We can use this to find the perpendicular height of the pyramid.

 

h=\sqrt{10^2-8^2}=6

 

We can substitute the values we have been given into the formula and work out the volume.

 

V=\cfrac{1}{3}\times 64 \times 6=128

 

So the volume is 128 \ cm^{3} .

5. The volume of this square based pyramid is 460 \ cm^{3}. Find the perpendicular height of the pyramid. Give your answer correct to 3 significant figures.

 

Volume of square based pyramid practice question 5

2.72 \ cm
GCSE Quiz False

2.73 \ cm
GCSE Quiz False

8.17 \ cm
GCSE Quiz True

8.16 \ cm
GCSE Quiz False

We can substitute the values we are given into volume of a pyramid formula and rearrange it to find the height, h.

 

\begin{aligned} V&=\cfrac{1}{3}Bh \\\\ 460&=\cfrac{1}{3}\times 13^2\times h \\\\ h&=\cfrac{460\times 3}{13^2} \\\\ h&=8.1656… \end{aligned}

 

So the perpendicular height is 8.17 \ cm^{3} (to 3 sf).

6. The volume of this square based pyramid is 8900 \ mm^{3}. Find the side length of the pyramid. Give your answer correct to 3 significant figures.

 

Volume of square based pyramid practice question 6

34.2 \ mm
GCSE Quiz False

342 \ mm
GCSE Quiz False

18.5 \ cm
GCSE Quiz False

18.5 \ mm
GCSE Quiz True

We can substitute the values we are given into the volume of a pyramid formula and rearrange it to find the base area.

 

The perpendicular height is given in cm, so needs converting to mm.

 

\begin{aligned} V&=\cfrac{1}{3}Bh \\\\ 8900&=\cfrac{1}{3}\times B\times 78 \\\\ B&=\cfrac{8900\times 3}{78}\\\\ B&=342.3076… \end{aligned}

 

The area of the base needs to be square rooted to find the side length.

 

\sqrt{342.3076…}=18.501…

 

So the perpendicular height is 18.5 \ mm (to 3 sf).

Volume of square based pyramid GCSE questions

1. The diagram shows a square based pyramid.

 

Volume of square based pyramid gcse question 1

 

Calculate the volume of the square based pyramid.

Give your answer correct to 3 significant figures

Give the correct units for your answer.

 

(4 marks)

Show answer

Area of the base is 120^{2}=14400 .

(1)

\cfrac{1}{3} \times 14400 \times 150

(1)

720 \ 000

(1)

m^3

(1)

2. The diagram shows a square based pyramid.

 

Volume of square based pyramid gcse question 2

 

Calculate the volume of the square based pyramid.

Give your answer correct to 3 significant figures

 

(4 marks)

Show answer

Area of the base is 32^{2}=1024 .

(1)

h=\sqrt{43^{2}-16^{2}}=39.912…

(1)

\cfrac{1}{3} \times 1024 \times 39.912…=13623.4…

(1)

13600 \ cm^3

(1)

3. The diagram shows a square based pyramid.

It is based on one of the pyramids of Giza.

 

Volume of square based pyramid gcse question 3

 

It has a volume of 2.6 million m^{3}.

It has a height of 147 \ m.

 

Find the side length, x of its square base.

Give your answer to 3 significant figures

 

(4 marks)

Show answer

2.6 million = 2 \ 600 \ 000

(1)

\cfrac{2 \ 600 \ 000 \times 3}{147}=53061.224…

(1)

\sqrt{53061.224…}=230.35…

(1)

230 \ m

(1)

4. The diagram shows a truncated square pyramid.

 

Volume of square based pyramid gcse question 4 image 1

 

Find the volume of the truncated square based pyramid.

Give your answer as an exact value.

 

(5 marks)

Show answer

Volume of square based pyramid gcse question 4 image 2

For using similar triangles to find the missing height.

(1)

For the height of large pyramid or the height of small pyramid, 15 \ cm and 5 \ cm.

(1)

For the volume of large pyramid or the volume of small pyramid, 720 or 26\cfrac{2}{3}.

(1)

720-26\cfrac{2}{3}

(1)

693\cfrac{1}{3} \ cm^3

(1)

Learning checklist

You have now learned how to:

  • Recognise a square based pyramid and its features
  • Calculate the volume of a square based pyramid
  • Solve problems involving volume
  • Find the total surface area of a square based pyramid

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