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Area of triangles and quadrilaterals

Substituting into formulaeThis topic is relevant for:

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.

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

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.**

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

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

DOWNLOAD FREEWork out the volume of the triangular prism:

**Write down the formula.**

Volume of prism = Area of cross section × depth

2**Calculate 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 .

3**Calculate the volume of the prism.**

The depth of the prism is 10cm .

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

4**Write 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

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 × width × depth

since the area of a rectangle is equal to height × width.

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

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

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 .

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.**

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 .

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 .

**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.

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

1. Work out the volume of the prism:

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}

2. Calculate the volume of the prism:

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}

3. Work out the volume of the prism:

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}

4. Work out the volume of the prism:

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}

5. The volume of this prism is 156cm^3 .

Work out the depth, x , of the prism.

13cm

1872cm

1.08cm

6.5cm

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

6. The volume of this prism is 270mm^3 . Work out the height, h , of the prism.

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}

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)**

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

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