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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}Bhwhere:
- 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.
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:
- Calculate the area of the base.
- Substitute values into the formula and solve.
- Write the answer, including the units.
How to calculate the volume of a pyramid
Volume and surface area of pyramids worksheet
Get your free volume of a pyramid worksheet of 20+ volume and surface area of pyramids questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE Volume and surface area of pyramids worksheet
Get your free volume of a pyramid worksheet of 20+ volume and surface area of pyramids questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEVolume of a pyramid examples
Example 1: calculating the volume with a diagram included
Calculate 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=492Substitute 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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:
Practice volume of a pyramid questions
1. Calculate the volume of the pyramid below.
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:
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.
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.
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:
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.
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)
(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)
(1)
3. A solid is made from a pyramid on top of a cuboid.
Calculate the volume of the compound 3D shape.
(3 marks)
(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
The next lessons are
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