Volume of a prism

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GCSE Maths Geometry and Measure 3D Shapes Prism

Volume Of A Prism

Here we will learn about the volume of a prism, including how to calculate the volume of a variety of prisms and how to find a missing length given the volume of a prism.

There are also volume and surface area of a prism 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 prism?

The volume of a prism is how much space there is inside a prism.

Imagine filling this L-shaped prism fully with water. The total amount of water inside the prism would represent the volume of the prism in cubic units.

To calculate the volume of a prism, we find the area of the cross section and multiply it by the depth.

Volume of prism = Area of cross section x depth

What is the volume of a prism?

How to calculate the volume of a prism

In order to calculate the volume of a prism:

Write down the formula.
Calculate the area of the cross section.
Calculate the volume of the prism.
Write the answer, including the units.

How to calculate the volume of a prism

Volume of a prism worksheet

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

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Volume of a prism worksheet

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

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Related lessons on prism shape

Volume of a prism is part of our series of lessons to support revision on prism shape. You may find it helpful to start with the main prism shape 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:

Volume of a prism examples

Example 1: volume of a triangular prism

Work out the volume of the triangular prism:

Write down the formula.

Volume of prism = Area of cross section × depth

2Calculate the area of the cross section.

\[\text{Area of triangle }=\frac{1}{2}\times{b}\times{h}\\ =\frac{1}{2}\times{8}\times{3}\\ =12\]

The area of the triangle is $12cm^2$ .

3Calculate the volume of the prism.

The depth of the prism is $10cm$ .

\[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =12\times{10}\\ =120\]

4Write the answer, including the units.

The measurements on this triangular prism are in centimetres so the volume will be measured in cubic centimetres.

Volume = $120cm^3$

Example 2: volume of a rectangular prism (cuboid)

A swimming pool is being built in the shape of a cuboid.

Calculate the volume of water in the pool when it is completely filled, in litres.

Write down the formula.

Volume of prism = Area of cross section × depth

Calculate the area of the cross section.

\[\text{Area of rectangle }=b\times{h}\\ =12\times{3}\\ =36\]

The area of the rectangle is $36m^2$ .

Calculate the volume of the prism.

The depth of the prism is $5m$ .

\[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =36\times{5}\\ =180\]

Write the answer, including the units.

The measurements on this prism are in metres so the volume will be measured in cubic metres, however we then need to convert this to litres, as it is stated in the question.

Volume = $180m^3$ .

$1m^3 = 1000L$ and so $180m^3 180,000L$ .

The volume of water in the swimming pool in litres is $180,000L$ .

Note: You may also use the formula:

Volume of cuboid $= height \times width \times depth$

since the area of a rectangle is equal to $height \times width.$

Example 3: volume of a hexagonal prism

Work out the volume of the prism:

Write down the formula.

Volume of prism = Area of cross section × depth

Calculate the area of the cross section.

In this example, we are told that the area of the hexagon is $50mm^2$ so we can move on to the next step.

Calculate the volume of the prism.

\[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =50\times{15}\\ =750\]

Write the answer, including the units.

The measurements on this prism are in millimetres so the volume will be measured in cubic millimetres.

Volume = $750mm^3$

Example 4: volume of a compound prism

Work out the volume of the L-shaped prism:

Write down the formula.

Volume of prism = Area of cross section × depth

Calculate the area of the cross section.

To calculate the area of the cross section we need to split it into two rectangles and work out the missing side lengths. We can then work out the area of each rectangle:

Rectangle A:

\[\text{Area }=7\times{4}\\ =28\]

Rectangle B:

\[\text{Area }=6\times{5}\\ =30\]

Total area:

28+30=58cm^2

Calculate the volume of the prism.

The depth of the prism is $12cm$ .

\[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =58\times{12}\\ =696\]

Write the answer, including the units.

The measurements on this prism are in cm so the volume will be measured in $cm^3$ .

Volume = $696cm^3$

Example 5: volume of a trapezoidal prism

Work out the volume of the prism:

Write down the formula.

Volume of prism = Area of cross section × depth

Calculate the area of the cross section.

\[\text{Area of cross section }=\frac{1}{2}(a+b)h\\ =\frac{1}{2}(2+4) \times 3\\ =9\mathrm{cm}^{2}\]

Calculate the volume of the prism.

\[\text{Volume of prism }= \text{Area of cross section } \times \text{ depth}\\ =9 \times 5\\ =45\]

Write the answer, including the units.

Volume = $45cm^3$ .

How to work out a missing length given the volume

Sometimes we might know the volume and some of the measurements of a prism and we might want to work out the other measurements. We can do this by substituting the values that we know into the formula for the volume of a prism and solving the equation that is formed.

Write down the formula.
Volume of a prism = Area of cross section × depth
Calculate the area of the cross section.
Substitute known values into the formula, and solve the equation.
Write the answer, including the units.

Missing length examples

Example 6: missing length in a pentagonal prism

The volume of this prism is $225cm^2$ . Work out the depth, L, of the prism:

Write down the formula.

Volume of prism = Area of cross section × depth

Calculate the area of the cross section.

In this example, we are told the area of the cross section is $25cm^2$

Substitute known values into the formula, and solve the equation.

\begin{aligned} \text{Volume of prism }&=\text{Area of cross section }\times \text{depth}\\ 225&=25 \times D\\ 25D&=225\\ D&=9 \end{aligned}

Write the answer, including the units.

Since the units in this question are in cm and $cm^3$ , the depth of the prism is $9cm$ .

Example 7: missing height in a trapezoidal prism

The volume of this prism is $336cm^3$ . Work out the height of the prism.

Write down the formula.

Volume of prism = Area of cross section × depth

Calculate the area of the cross section.

\[\text{Area of trapezium }=\frac{1}{2}(a+b)h\\ =\frac{1}{2}(5+9) \times h\\ =7h\]

Substitute known values into the formula, and solve the equation.

\[\text{Volume of prism }=\text{Area of cross section } \times \text{depth}\\ 336=7h \times 8\\ 336=56h\\ 56h=336\\ h=6\]

Write the answer, including the units.

Since the units in this question are in cm and $cm^3$ , the height of the prism is $6cm$ .

Common misconceptions

Missing/incorrect units

You should always include units in your answer. Remember, volume is measured in units cubed (e.g. $mm^3, cm^3, m^3$ etc)

Calculating with different units

You need to make sure all measurements are in the same units before calculating volume. (E.g. you can’t have some in cm and some in $m$ )

Using the incorrect formula

Be careful to apply the correct prism related formula to the correct question type.

Practice volume of a prism questions

192cm^3

48cm^3

96cm^3

40cm^3

\text{Area of triangle }=\frac{1}{2} \times 4 \times 6\\ =12\mathrm{cm}^{2}

\text{Volume of triangular prism }=12 \times 8\\ =96\mathrm{cm}^{3}

220cm^3

308cm^3

1540cm^3

264cm^3

\text{Area of trapezium }=\frac{1}{2}(5+7) \times 4\\ =24\mathrm{cm}^{2}

\text{Volume of prism }=24 \times 11\\ =264\mathrm{cm}^{3}

240cm^3

2.4cm^3

2400cm^3

5760cm^3

Area of cross section = $24cm^2$

\text{Volume of prism }=24 \times 10\\ =240 \mathrm{cm}^{3}

84cm^3

1680cm^3

1008cm^3

1344cm^3

Area of triangle $A$ :

\text{Area }=\frac{1}{2} \times 7 \times 6\\ =21\mathrm{cm}^{2}

Area of rectangle $B$ :

\text{Area }=7 \times 9\\ =63 \mathrm{cm}^{2}

\text{Total area: } 21+63=84\mathrm{cm}^{2}

\text{Volume of prism }=84 \times 16\\ =1344\mathrm{cm}^{3}

13cm

1872cm

1.08cm

6.5cm

\text{Volume of prism} = \text{Area of cross section} \times \text{depth}\\ 156=12x\\ 12x=156\\ x=13

150mm

30mm

6mm

60750mm

\text{Area of parallelogram }=5 \times h \begin{aligned} \text{Volume of prism }&=\text{Area of cross section }\times \text{depth}\\ 270&=5h \times 9\\ 270&=45h\\ 45h&=270\\ h&=6 \end{aligned}

Volume of a prism GCSE questions

1. Work out the volume of the prism. State the units in your solution.

(3 marks)

Show answer

$\text{Area of cross section }=2 \times 4 + 4 \times 1=12\text{cm}^2$
or
$\text{Area of cross section }=2 \times 3 + 6 \times 1=12\text{cm}^2$
For calculating the cross-sectional area of the prism

(1)

$\text{Volume of prism: }12 \times 5=60$
For calculating the volume of the prism

(1)

$60cm^3$
For correct units

(1)

2. The volume of the cuboid is twice the volume of the triangular prism. Work out the height, $y$ , of the cuboid.

(5 marks)

Show answer

$\frac{1}{2} \times 9 \times 4=18\mathrm{cm}^{2}$
For area of the cross section (triangle)

(1)

$18 \times 5=90 \mathrm{cm}^{3}$
For volume of the triangular prism

(1)

$3 \times y \times 6=18y$
For volume of the cuboid

(1)

$90 \times 2=180\text{ and } 18y=180$
Forming an equation to calculate the height of the cuboid

(1)

$y=10cm$
For the correct answer

(1)

3. (a) Calculate the volume of the trapezoidal prism.

(b) The prism is made from aluminum, which has a density of $2.7g/cm^3$ . Work out the mass of the prism. State the units in your answer.

(4 marks)

Show answer

(a)
$\frac{1}{2}(2+6) \times 4 = 16\mathrm{cm}^{2}$
For area of the cross section (trapezium)

(1)

$16 \times 8=128 \mathrm{cm}^{3}$
For volume of the prism

(1)

(b)
$128 \times 2.7$
For using Mass = Density $\times$ Volume

(1)

$=345.6g$
For correct solution including units

(1)

Learning checklist

You have now learned how to:

Know and apply formulae to calculate the volume of prisms

Use the properties of faces, surfaces, edges and vertices to solve problems in 3-D

Calculate the volume of composite solids

The next lessons are

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