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.

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

katex is not defined.

The pyramid height should be perpendicular to its base.

The formula can also be written as

where

• katex is not defined represents the volume of the pyramid,
• katex is not defined represents the area of its base,
• katex is not defined 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.

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?

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

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.

1. Calculate the area of the base.

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

2Substitute values into the formula and solve.

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

3Write the answer, including the units.

The volume is katex is not defined

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

Find the volume of the square based pyramid.

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.

Substitute values into the formula and solve.

Show step

The volume of a square based pyramid can be found by using the formula. But the perpendicular height is not given. It can be found using Pythagoras’ theorem. The slant height of the pyramid katex is not defined is the hypotenuse.

katex is not defined

katex is not defined

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.

Substitute values into the formula and solve.

Show step

The volume of a square based pyramid can be found by using the formula. But we do not know the perpendicular height. We have been given the lateral edge length. We will need to apply Pythagoras’ theorem twice to find the perpendicular height.

The value katex is not defined comes from using the midpoint of the base edge to make right angled triangles.

First we find the slant length from the lateral edge length.

katex is not defined

Then we can find the perpendicular height of the square based pyramid.

katex is not defined

katex is not defined

Example 5: calculating the height given the volume

The volume of the square based pyramid is katex is not defined Find the perpendicular height, katex is not defined Give your answer to katex is not defined significant figures.

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

The volume of the square based pyramid is katex is not defined Find the base edge. Give your answer correct to katex is not defined decimal place.

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 katex is not defined or katex is not defined Square units such as katex is not defined or katex is not defined 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.

katex is not defined

Or

katex is not defined

• 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.

Related lessons

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:

• Pyramid
• Surface area of a pyramid
• Triangular based pyramid
• Square based pyramid
• Volume of a pyramid

The next lessons are

• Cylinder
• Pythagoras theorem
• Trigonometry

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