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3D shapes

Area of a triangle

Substituting into formulaeTotal surface area

This topic is relevant for:

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.

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).

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.**

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

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

COMING SOONWork out the volume of this triangular prism

**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 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

3**Work out the calculation.**

*Volume of triangular prism = * 3 × 7

*Volume of triangular prism = * 21

4**Write 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

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 × 18

**Work out the calculation.**

*Volume of triangular prism = * 20 × 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

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 × 21

**Work out the calculation.**

*Volume of triangular prism =* 50 × 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

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.

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**.

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

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*

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

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

**Solve the equation.**

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

**Write the answer, including the units.**

x=2.5mm

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*

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

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

**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:

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 .

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)**

**(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)**

**(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)**

**(1)**

**(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)**

**(1)**

(b)

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

**(1)**

**(1)**

**(1)**

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

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