# 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).

## How to calculate the volume of a triangular prism

In order to calculate the volume of a triangular prism:

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

## Volume of a triangular prism examples

### Example 1: volume of a triangular prism

Work out the volume of this triangular prism

1. 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 × 7

3Work out the calculation.

Volume of triangular prism = 3 × 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

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

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

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

Volume of triangular prism = 20 × 18

Volume of triangular prism = 20 × 18

Volume of triangular prism = 360

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

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

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 × 21

Volume of triangular prism = 50 × 21

Volume of triangular prism = 1050

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:

1. Write down the formula.
Volume of a triangular prism = area of triangular cross section x length
2. Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.
3. Solve the equation.
4. 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.

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

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

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

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

x=8cm

### Example 5: finding a missing height

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

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

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

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

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

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 .

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

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

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}

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

y=11mm

### Common misconceptions

• Missing/incorrect units

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

1. Work out the volume of the triangular prism

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}

2. Work out the volume of the triangular prism

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}

3. Work out the volume of the triangular prism

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}

4. The volume of this triangular prism is 84cm^3 . Work out the length, x , of the triangular prism

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 .

5. The volume of this triangular prism is 405m^3 . Work out the height of the triangular prism

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 .

6. The volume of this triangular prism is 45cm^3 . Work out the length of y .

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)

\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)

\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)

(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

## Still stuck?

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