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3D shapes Area of a triangle Substituting into formulae How to calculate surface area

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GCSE Maths Geometry and Measure 3D Shapes Triangular Prism

Volume

Volume Of A Triangular Prism

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

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

The volume of a triangular prism is how much space there is inside a triangular prism. A triangular prism is a polyhedron (3D shape made from polygons) with two congruent triangular ends connected by three rectangles.

To work this out we find the area of the triangular cross-section and multiply it by the length.

Volume of a triangular prism = Area of triangular cross section x length

E.g.

\[\begin{array}{l} \text{Area of triangular cross-section:}\\ \text{Area }=\frac{1}{2}bh\\ \text{Area }=\frac{1}{2} \times 4 \times 5\\ \text{Area }=10\mathrm{cm}^{2}\\ \\ \text{Volume of triangular prism:}\\ \text{Volume }= \text{Area of triangular cross-section } \times \text{ length}\\ \text{Volume }=10 \times 11\\ \text{Volume }=110\mathrm{cm}^{3} \end{array}\]

Volume is measured in cubic units (e.g. $mm^3, cm^3, m^3$ etc).

What is volume of a triangular prism?

How to calculate the volume of a triangular prism

In order to calculate the volume of a triangular prism:

Write down the formula.
Volume of a triangular prism = Area of triangular cross section $\times$ length
Calculate the area of the triangular cross-section and substitute the values.
Work out the calculation.
Write the answer, including the units.

How to calculate the volume of a triangular prism

Volume and surface area of a triangular prism worksheet

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

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Volume and surface area of a triangular prism worksheet

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

DOWNLOAD FREE

Volume of a triangular prism examples

Example 1: volume of a triangular prism

Work out the volume of this triangular prism

Write down the formula.

Volume of a triangular prism = Area of triangular cross section x length

2Calculate the area of the triangular cross-section and substitute the values.

The base of the triangle is $2cm$ and the height of the triangle is $3cm$ .

\[\begin{array}{l} \text{Area of triangle }=\frac{1}{2} \times b \times h\\ \text{Area of triangle }=\frac{1}{2} \times 2 \times 3\\ \text{Area of triangle }=3 \end{array}\]

The area of the triangle is $3cm^2$ .

The length of the prism is $7cm$ .

Volume of triangular prism = Area of triangular cross section x length

Volume of triangular prism = $3 \times 7$

3Work out the calculation.

Volume of triangular prism = $3 \times 7$

Volume of triangular prism = $21$

4Write the answer, including the units.

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

Volume = $21cm^3$

Example 2: triangular prism with isosceles triangle

Work out the volume of the triangular prism

Write down the formula.

Volume of a triangular prism = Area of triangular cross section x length

Calculate the area of the triangular cross-section and substitute the values.

\[\begin{array}{l} \text{Area of triangle }=\frac{1}{2} \times b \times h\\ \text{Area of triangle }=\frac{1}{2} \times 5 \times 8\\ \text{Area of triangle }=20 \end{array}\]

Volume of triangular prism = Area of triangular cross section x length

Volume of triangular prism = $20 \times 18$

Work out the calculation.

Volume of triangular prism = $20 \times 18$

Volume of triangular prism = $360$

Write the answer, including the units.

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

Volume = $360mm^3$

Example 3: different units

Work out the volume of this triangular prism

Write down the formula.

Volume of a triangular prism = Area of triangular cross section x length

Calculate the area of the triangular cross-section and substitute the values.

This time, the triangular prism is the other way up so we start by calculating the area of the base. There are some measurements in both $m$ and $cm$ here so we need to make the units the same before we begin calculating. The easiest thing to do in this example is to convert $0.1m$ to $10cm$ .

\[\begin{array}{l} \text{Area of triangle }=\frac{1}{2} \times b \times h\\ \text{Area of triangle }=\frac{1}{2} \times 10 \times 10\\ \text{Area of triangle }=50 \end{array}\]

Since the triangular prism is the other way up, the length that we need to multiply by is the height of the prism, $21cm$ .

Volume of triangular prism = Area of triangular cross section x length

Volume of triangular prism = $50 \times 21$

Work out the calculation.

Volume of triangular prism = $50 \times 21$

Volume of triangular prism = $1050$

Write the answer, including the units.

The measurements that we used for this triangular prism are in $cm$ so the volume will be measured in $cm^3$ .

Volume = $1050cm^3$

Calculating a missing length

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

How to work out a missing length given the volume

In order to calculate the length given the volume:

Write down the formula.
Volume of a triangular prism = area of triangular cross section x length
Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.
Solve the equation.
Write the answer, include the units.

Missing length examples

Example 4: finding a missing length

The volume of this triangular prism is $168cm^3$ . Work out the length, $x$ , of the triangular prism.

Write down the formula.

Volume of a triangular prism = area of triangular cross section x length

Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.

\[\begin{array}{l} \text{Area of triangle }=\frac{1}{2} \times b \times h\\ \text{Area of triangle }=\frac{1}{2} \times 7 \times 6\\ \text{Area of triangle }=21 \end{array}\]

\[\begin{aligned} \text{Volume of a triangular prism } &= \text{ area of triangular cross-section } \times { length}\\ 168&=21 \times x \end{aligned}\]

Solve the equation.

\[\begin{aligned} 21x&=168\\ x&=8 \end{aligned}\]

Write the answer, including the units.

$x=8cm$

Example 5: finding a missing height

The volume of this triangular prism is $80mm^3$ . Work out the height of the prism.

Write down the formula.

Volume of a triangular prism = area of triangular cross section x length

Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.

\[\begin{array}{l} \text{Area of triangle }=\frac{1}{2} \times b \times h\\ \text{Area of triangle }=\frac{1}{2} \times 4 \times h\\ \text{Area of triangle }=2h \end{array}\]

\[\begin{aligned} \text{Volume of a triangular prism } &= \text{ area of triangular cross-section } \times { length}\\ 80&=2h \times 16\\ 80 &= 32h \end{aligned}\]

Solve the equation.

\[\begin{aligned} 32h&=80\\ h&=2.5 \end{aligned}\]

Write the answer, including the units.

$x=2.5mm$

Example 6: finding a missing base, different units

The volume of this triangular prism is $440mm^2$ . Work out the length labelled $y$ .

Write down the formula.

Volume of a triangular prism = area of triangular cross section x length

Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.

\[\begin{array}{l} \text{Area of triangle }=\frac{1}{2} \times b \times h\\ \text{Area of triangle }=\frac{1}{2} \times y \times 4\\ \text{Area of triangle }=2y \end{array}\]

Notice here that we need to work in mm since the volume is in $mm^3$ . Therefore we need to convert $2cm$ to $20mm$ .

\[\begin{aligned} \text{Volume of a triangular prism } &= \text{ area of triangular cross-section } \times { length}\\ 440&=2y \times 20\\ 440 &= 40y \end{aligned}\]

Solve the equation.

\[\begin{aligned} 40y&=440\\ y&=11 \end{aligned}\]

Write the answer, including the units.

$y=11mm$

Common misconceptions

Missing/incorrect units

You should always include units in your answer.
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 wrong formula

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

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

110 \mathrm{cm}^{3}

55 \mathrm{cm}^{3}

240 \mathrm{cm}^{3}

120 \mathrm{cm}^{3}

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 3 \times 8\\ &=12\mathrm{cm}^{2} \end{aligned}

\begin{aligned} \text{Volume of triangular prism }&=12 \times 10\\ &=120\mathrm{cm}^{3} \end{aligned}

252 \mathrm{cm}^{3}

42 \mathrm{cm}^{3}

1764 \mathrm{cm}^{3}

126 \mathrm{cm}^{3}

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 6 \times 6\\ &=18\mathrm{cm}^{2} \end{aligned}

\begin{aligned} \text{Volume of triangular prism }&=18 \times 7\\ &=126\mathrm{cm}^{3} \end{aligned}

924 \mathrm{mm}^{3}

9.24 \mathrm{cm}^{3}

92.4 \mathrm{mm}^{3}

9240 \mathrm{mm}^{3}

Notice that one of the measurements is in $mm$ and the others are in $cm$ . We can change $1.4cm$ to $14mm$ and $2.2cm$ to $22mm$ .

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 14 \times 6\\ &=42\mathrm{mm}^{2} \end{aligned}

\begin{aligned} \text{Volume of triangular prism }&=42 \times 22\\ &=924\mathrm{mm}^{3} \end{aligned}

3.5cm

2016cm

7cm

1008cm

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 3 \times 8\\ &=24\mathrm{cm}^{2} \end{aligned}

\begin{aligned} \text{Volume of triangular prism }&=12 \times x\\ 84 &= 12x\\ 7&=x \end{aligned}

The length is $7cm$ .

45m

9m

2.5m

4.5m

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 6 \times h\\ &=3h \end{aligned}

\begin{aligned} \text{Volume of triangular prism }&=3h \times 15\\ 405 &= 45h\\ 9&=h \end{aligned}

The height is $9m$ .

5cm

0.25cm

0.5cm

4050cm

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 4 \times y\\ &=4y \end{aligned}

Notice that the height of the triangular prism is in mm however the volume is in cm3. Therefore we need to change $45mm$ to $4.5cm$ .

\begin{aligned} \text{Volume of triangular prism }&=2y \times 4.5\\ 45 &= 9h\\ 5&=h \end{aligned}

The length of $y$ is $5cm$ .

Volume of a triangular prism GCSE questions

1. Work out the volume of the triangular prism.

(2 marks)

Show answer

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 5 \times 2\\ &=5 \mathrm{cm}^{2} \end{aligned}

(1)

\begin{aligned} \text{Volume of triangular prism }&=5 \times 6\\ &=30\mathrm{cm}^{3} \end{aligned}

(1)

2. These triangular prisms have the same volume. Work out the height, $h$ , of prism $B$ .

(5 marks)

Show answer

\begin{aligned} \text{Area of triangle A }&=\frac{1}{2} \times 8 \times 3\\ &=12 \mathrm{cm}^{2} \end{aligned}

(1)

\begin{aligned} \text{Volume of triangular prism A }&=12 \times 12\\ &=144\mathrm{cm}^{3} \end{aligned}

(1)

$\begin{aligned} \text{Area of triangle B }&=\frac{1}{2} \times 4 \times h\\ &=2h \end{aligned}$
$\begin{aligned} \text{Volume of triangular prism B}&=2h \times 14.4\\ \end{aligned}$

(1)

144 = 28.8h

(1)

h = 5cm

(1)

3. (a) Work out the volume of the triangular prism.

(b) A section, $4cm$ tall, is cut off of the top of the triangular prism. Find the volume of the remaining shape.

(5 marks)

Show answer

(a)
$\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 5 \times 8\\ &=20 \mathrm{cm}^{2} \end{aligned}$

(1)

\begin{aligned} \text{Volume of triangular prism }&=20 \times 15\\ &=300\mathrm{cm}^{3} \end{aligned}

(1)

(b)
$\begin{aligned} \text{Area of small triangle }&=\frac{1}{2} \times 2.5 \times 4\\ &=5 \mathrm{cm}^{2} \end{aligned}$

(1)

\begin{aligned} \text{Volume of small triangular prism }&=5 \times 15\\ &=75\mathrm{cm}^{3} \end{aligned}

(1)

\begin{aligned} \text{Volume of remaining shape }&=300-75\\ &=225\mathrm{cm}^{3} \end{aligned}

(1)

Learning checklist

You have now learned how to:

Know and apply formula to calculate the volume of prisms

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

The next lessons are

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