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3D shapes

Area of a rectangle Substituting into formulae DecimalsThis topic is relevant for:

Here we will learn about the volume of a cuboid, including how to calculate the volume of a cuboid and how to find missing lengths of a cuboid given its volume.

There are also volume and surface area of a cuboid* *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 cuboid** is the amount of space there is within a cuboid.

**Cuboids **are 3 dimensional shapes with 6 rectangular faces.

The formula for the volume of a cuboid is:

Volume = length \times width \times heightVolume is measured in cubic units, e.g. mm^3, cm^3 or m^3.

This cuboid is made from 24 unit cubes.

Its volume is

\begin{aligned} &Volume = length \times width \times height \\ &Volume = 2 \times 4 \times3 \\ &Volume = 24 \; cm^3 \end{aligned}The units of volume are cubed units

E.g. mm^3 (cubic millimetres), cm^3 (cubic centimetres), m^3 (cubic metres) .

In order to calculate the volume of a cuboid:

**Write down the formula.**

Volume = length \times width \times height

2**Substitute the values into the formula. **

Make sure the units are the same for all measurements

3**Substitute the values into the formula. **

4**Write the answer, include the units.**

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

DOWNLOAD FREEGet your free volume of a cuboid worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEWork out the volume of the cuboid:

**Write down the formula.**

2**Substitute the values into the formula.**

Here the length is 10 \; cm , the width is 2 \; cm and the height is 5 \; cm .

Volume = 2 \times 10 \times 53**Work out the calculation.**

4**Write the answer, include the units.**

The measurements are in cm therefore the volume will be in cm^3.

Volume = 100cm^3Work out the volume of this cube:

**Write down the formula.**

Volume = length \times width \times height

**Substitute the values into the formula.**

Since this is a cube, the length of the cuboid, width of the cuboid and height of the cuboid are all 6 \; cm.

Volume = 6 \times 6 \times 6

**Work out the calculation.**

\begin{aligned} &Volume = 6 \times 6 \times 6 \\\\ &Volume = 216 \end{aligned}

**Write the answer, include the units.**

Volume=216 \; cm^3

Work out the volume of this cuboid:

**Write down the formula.**

Volume = length \times width \times height

**Substitute the values into the formula.**

Notice here that one of the units is in cm whilst the others are in m. We need all the units to be the same to calculate the volume.

We can change cm to m: 50cm = 0.5m.

Now that we have all of the measurements in m, we can calculate the volume:

Volume = 4 \times 2 \times 0.5

**Work out the calculation.**

\begin{aligned} &Volume = 4 \times 2 \times 0.5 \\\\ &Volume = 4 \end{aligned}

**Write the answer, include the units.**

Since the measurements that we used were in metres, the volume will be in cubic metres.

Volume=4 \; m^3

Sometimes we might know the volume and some of the measurements of a cuboid 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 cuboid and solving the equation that is formed.

**Write down the formula.**

Volume = length \times width \times height

2**Substitute the values that you do know into the formula.**

Make sure the units are the same for all measurements

3**Solve the equation.**

4**Write the answer, include the units.**

The volume of the cuboid is 56 \; cm^3 . Work out the width of the cuboid.

**Write down the formula.**

Volume = length \times width \times height

**Substitute the values that you do know into the formula.**

In this example, we know that the volume of the cuboid is 56 cubic centimetres, the width is 7 \; cm and the height is 2 \; cm.

Substituting these into the formula:

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
56&=7 \times w \times 2\\\\
56&=14w
\end{aligned}

**Solve the equation.**

\begin{aligned}
14w&= 56\\\\
w&=4
\end{aligned}

**Write the answer, include the units.**

Since the measurements in this question were in cm and cm^3 , the width will be in cm

w = 4 \;cm

The cuboid below has a square base. The height of the cuboid is 8 \; m and the volume of the cuboid is 32 \; m^3 . Find the length of the cuboid.

**Write down the formula.**

Volume = length \times width \times height

**Substitute the values that you do know into the formula.**

In this example, we know that the volume of the cuboid is 32 cubic metres and the height is 8 \; cm .

Substituting these into the formula:

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
32&=l \times w \times 8\\\\
32&=8lw
\end{aligned}

**Solve the equation.**

\begin{aligned} 8lw&=32\\\\ lw&=4 \end{aligned}

Since the base is a square, we know that the length and width are the same.

Therefore we are looking for a number that, when multiplied by itself, makes 4. We need to find the square root of 4.

\sqrt{4}=2

The length and width of this cuboid are 2.

**Write the answer, include the units.**

l=2 \;m

Work out the dimensions of a cube which has a volume of 64 \; cm^3

**Write down the formula.**

Volume = length \times width \times height

**Substitute the values that you do know into the formula.**

The only value we currently know is the volume is 64 cubic cm.

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
64&=l \times w \times h
\end{aligned}

**Solve the equation.**

lwh=64

Since the shape is a cube, we know that the length, width and height are all the same. Therefore we are looking for a number that, when multiplied by itself three times, makes 64. We need to find the cube root of 64.

\sqrt[3]{64}=4

The length, width and height of this cube are 4.

**Write the answer, include the units.**

The cube is 4cm \times 4cm \times 4cm

**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

**Dividing by three rather than cube rooting**

If you are given the volume of a cube and you need to find the side length, remember the inverse of cubed is cube root, not divide by 3.

E.g. if the volume of a cube is 8cm^3 , the side length is

\sqrt[3]{8}=2 \; \mathrm{cm} (not 8\div3 )

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

144 \; \mathrm{cm}^{2}

12 \; \mathrm{cm}^{3}

144 \; \mathrm{cm}^{3}

36 \; \mathrm{cm}^{3}

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
\text{Volume }&= 12 \times 3 \times 4\\\\
\text{Volume }&=144 \; \mathrm{cm}^{3}
\end{aligned}

2. Work out the volume of the cube

64 \; \mathrm{m}^{3}

512 \;\mathrm{m}^{3}

8 \;\mathrm{m}^{3}

24 \; \mathrm{m}^{3}

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
\text{Volume }&= 8 \times 8 \times 8\\\\
\text{Volume }&=512 \; \mathrm{m}^{3}
\end{aligned}

3. Work out the volume of this cuboid

1748 \; \mathrm{cm}^{3}

174.8 \; \mathrm{m}^{3}

17.48 \; \mathrm{m}^{3}

17.48 \; \mathrm{cm}^{3}

Here there are measurements in cm and m so we must make the units the same before calculating the volume.

We can convert 380 \; cm to 3.8 \;m.

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ \text{Volume }&= 2.3 \times 2 \times 3.8\\\\ \text{Volume }&=17.48 \; \mathrm{m}^{3} \end{aligned}

Since the measurements we used were in metres, the volume will be in cubic metres.

4. The volume of this cuboid is 600 \; cm^{3} .

Work out the height of the cuboid.

3 \; cm

24 \; cm

200 \; cm

75 \; cm

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
600&= 25 \times 8 \times h\\\\
600&=200h\\\\
h&=3\; \mathrm{cm}
\end{aligned}

5. The base of this cuboid is a square. The volume of the cuboid is 450\; cm^{3} .

Work out the height of the cuboid.

45 \; cm

4.5 \; cm

2.1 \; cm

6.7 \; cm

Since the base of the cuboid is a square, the length and the width are both 10 \; cm.

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ 450&= 25 \times 10 \times h\\\\ 450&=100h\\\\ h&=4.5 \; \mathrm{cm} \end{aligned}

6. The volume of this cube is 343 \; cm^{3} .

What is the length of the cube?

7 \; cm

114.3 \; cm

18.5 \; cm

57.2 \; cm

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
343&=l \times w \times h
\end{aligned}

Since this is a cube we know that the length, width and height are all equal. Therefore we are looking for a number that, when multiplied by itself 3 times, makes 343. We need the cube root of 343.

\sqrt[3]{343}=7

The length, width and height of the cube are all 7 \;cm.

1. Work out the volume of this cuboid. Give your answer in cm^3.

**(3 marks)**

Show answer

0.9 \; m = 90 \;cm

**(1)**

\text{Volume }=90 \times 40 \times 20

**(1)**

\text{Volume }=72000 \; \mathrm{cm}^{3}

**(1)**

2. A paddling pool is in the shape of a cuboid.

(a) Work out the volume of the paddling pool.

(b) Sam wants to fill the pool so that it is \frac{5}{6} full.

Water flows out of her hose pipe and into the pool at a rate of 20 litres per minute.

Given that 1\;\mathrm{m}^{3}=1000 \;\mathrm{l} , calculate the length of time it would take Sam to fill the pool so that it is \frac{5}{6} full. Give your answer in hours.

**(6 marks)**

Show answer

(a)

\text{Volume }=4 \times 1.8 \times 0.6

**(1)**

\text{Volume }=4.32 \; \mathrm{m}^{3}

**(1)**

(b)

4.32 \times 1000=4320 \; \mathrm{l}

**(1)**

**(1)**

**(1)**

**(1)**

3. A carton of orange juice is shown below. The carton is completely full. The orange juice is poured into another container, as shown below.

What will the height of the orange juice in the container be?

**(5 marks)**

Show answer

\text{Volume of orange juice: }6 \times 10 \times 30

**(1)**

\text{Volume of orange juice: }1800 \; \mathrm{cm}^{3}

**(1)**

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ 1800&=40 \times 9 \times h \end{aligned}

**(1)**

1800 = 360 \;h

**(1)**

h = 5 \; cm

**(1)**

You have now learned how to:

- Know and apply the formula to calculate the volume of cuboids
- Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3-D

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