Volume of a Prism

What you'll learn

How to spot the cross-section of a prism.
How to use volume = area of cross-section × length.
How to handle triangular, trapezium and L-shaped prisms.
How to use unit conversions for litres, buckets, boxes and containers.

The basic idea

Before volume, you need to be confident with area. Area measures flat space in a 2D shape, using square units like cm².

Volume measures 3D space, using cubic units like cm³.

Definition

Volume, prism and cross-section

Volume is the amount of space inside or taken up by a 3D solid.
A prism is a 3D solid with the same 2D shape repeated all the way through.
The cross-section is the repeated 2D shape you see if you slice straight through the prism.

The length or depth of the prism is how far the cross-section has been stretched.

A prism is made by stretching the same cross-section through a length or depth.

Key Idea

Main formula

For any prism, use $\times L$ , where $A$ is the area of the cross-section and $L$ is the length of the prism.

Example

A rectangular prism

A prism has a rectangular cross-section measuring 6 cm by 5 cm. The prism is 8 cm long. Find its volume.

Rectangular prism with a 6 cm by 5 cm rectangular cross-section stretched 8 cm long.

Find the area of the rectangular cross-section.
$\times 5 = 30\text{ cm}^2$
Multiply by the length of the prism.
$\times 8 = 240\text{ cm}^3$
The volume is 240 cm³.

Composite cross-sections

A composite shape is made by joining simpler shapes together. For an L-shaped cross-section, split it into rectangles, find each area, then add them.

Example

L-shaped prism

An L-shaped cross-section can be split into two non-overlapping rectangles: one is 5 cm by 2 cm, and the other is 3 cm by 4 cm. The prism is 10 cm deep. Work out the volume.

L-shaped prism cross-section split into two rectangles before being extended 10 cm deep.

Find the area of the first rectangle.
$cm2A_1 = 5 \times 2 = 10\text{ cm}^2$
Find the area of the second rectangle.
$cm2A_2 = 3 \times 4 = 12\text{ cm}^2$
Add the areas to get the cross-section area.
$22\text{ cm}^2$
Multiply by the depth of the prism.
$\times 10 = 220\text{ cm}^3$
The volume is 220 cm³.

Common Mistake

Adding the depth

Do not add the prism length to the cross-section area. Area is in cm² and length is in cm, so you multiply them to make cm³.

Triangular prisms

A triangular prism has a triangle as its cross-section. A right-angled triangle contains a 90° angle.

For a triangle:

\frac{1}{2}bh

where $b$ is the base and $h$ is the perpendicular height.

Example

Right-angled triangular prism

A triangular prism has a right-angled triangular cross-section with perpendicular sides 6 cm and 8 cm. The prism is 9 cm long. Find the volume.

Right-angled triangular prism with perpendicular sides 6 cm and 8 cm and length 9 cm.

Find the area of the triangular cross-section.
$\frac{1}{2} \times 6 \times 8 = 24\text{ cm}^2$
Multiply by the length of the prism.
$\times 9 = 216\text{ cm}^3$
The volume is 216 cm³.

Common Mistake

Using the hypotenuse

In a right-angled triangle, the hypotenuse is the longest side opposite the right angle. It is not needed for triangle area unless it is paired with a perpendicular height.

The hypotenuse is opposite the right angle and is not one of the perpendicular base-height pair.

Trapezium prisms

A trapezium is a four-sided shape with one pair of parallel sides. Parallel sides are always the same distance apart and never meet.

For a trapezium:

\frac{1}{2}(a+b)h

where $a$ and $b$ are the parallel sides, and $h$ is the perpendicular height between them.

Example

Trapezium cross-section

A prism has a trapezium cross-section. The parallel sides are 6 cm and 10 cm, and the perpendicular height is 5 cm. The prism is 11 cm long. Find the volume.

Trapezium prism with parallel sides 6 cm and 10 cm, perpendicular height 5 cm, and length 11 cm.

Find the area of the trapezium.
$\frac{1}{2}(6+10)\times 5 = 40\text{ cm}^2$
Multiply by the length of the prism.
$\times 11 = 440\text{ cm}^3$
The volume is 440 cm³.

Working backwards

Sometimes the question gives you the volume and asks for a missing length, often called $x$ .

Use the volume formula, then solve the equation.

Example

Finding a missing height

A triangular prism has a right-angled triangular cross-section with base 7 cm and height $x$ cm. The prism is 12 cm long. Its volume is 252 cm³. Find $x$ .

Right-angled triangular prism where the perpendicular height of the cross-section is unknown.

Write the volume equation.
$12×7×x×12=252\frac{1}{2}\times 7 \times x \times 12 = 252$
Simplify the numbers.
$42 x = 252$
Divide both sides by 42.
$\frac{252}{42} = 6$
The missing height is 6 cm.

Tip

Keep the formula visible

When working backwards, write the full volume formula first. This helps you avoid forgetting the $12\frac{1}{2}$ in triangle questions.

Units, litres and containers

For containers, volume often connects to capacity.

1 ml = 1 cm³
1 litre = 1000 cm³

Example

Filling a container with buckets

A rectangular container is 70 cm long, 50 cm wide and 40 cm high. Each bucket holds 12 litres. How many buckets are needed to fill the container?

Find the volume of the container.
$\times 50 \times 40 = 140000\text{ cm}^3$
Convert cm³ to litres.
$litres140000\text{ cm}^3 = 140\text{ litres}$
Divide by the amount in one bucket.
$140 \div 12 = 11.666\ldots$
You need 12 buckets, because 11 buckets would not completely fill the container.

Tip

Rounding in capacity questions

If you are filling a container, round up. If you are finding how many cups can be completely filled, round down.

Boxes inside a larger cuboid

For packing boxes into a van or container, check how many fit along each direction.

Example

Packing boxes

A van space is 2.4 m long, 1.5 m wide and 1.2 m high. Each box is 60 cm long, 30 cm wide and 40 cm high. How many boxes fit?

Packing identical boxes into a cuboid van space requires checking how many fit along each direction.

Convert the van dimensions to centimetres: 2.4 m is 240 cm, 1.5 m is 150 cm, and 1.2 m is 120 cm.
Find how many boxes fit along the length.
$240 \div 60 = 4$
Find how many fit along the width and height.
$150 \div 30 = 5,\qquad 120 \div 40 = 3$
Multiply the three amounts.
$\times 5 \times 3 = 60$
The van can fit 60 boxes.

Cubes and surface area

A cube is a cuboid where every edge has the same length. The surface area is the total area of all the outside faces.

A cube has 6 identical square faces.

Example

Volume from surface area

The total surface area of a cube is 294 cm². Find its volume.

Divide by 6 to find the area of one square face.
$294 \div 6 = 49\text{ cm}^2$
Find the side length of the square face.
$49=7\sqrt{49}=7$
Cube the side length to find the volume.
$7^3 = 343\text{ cm}^3$
The volume is 343 cm³.

Exam technique

In the exam

Identify the cross-section first, then find its area.
Multiply the cross-section area by the prism length or depth.
Check units carefully, especially metres to centimetres and cm³ to litres.
For missing lengths, set up an equation using the volume formula before solving.

Self review

Check yourself

Can you point to the cross-section and the length of the prism in a diagram?
Which area formula would you use for a triangle or trapezium cross-section?
When should you round up, and when should you round down, in container questions?

Volume of a Prism

What you'll learn

How to spot the cross-section of a prism.
How to use volume = area of cross-section × length.
How to handle triangular, trapezium and L-shaped prisms.
How to use unit conversions for litres, buckets, boxes and containers.

The basic idea

Before volume, you need to be confident with area. Area measures flat space in a 2D shape, using square units like cm².

Volume measures 3D space, using cubic units like cm³.

Definition

Volume, prism and cross-section

Volume is the amount of space inside or taken up by a 3D solid.
A prism is a 3D solid with the same 2D shape repeated all the way through.
The cross-section is the repeated 2D shape you see if you slice straight through the prism.

The length or depth of the prism is how far the cross-section has been stretched.

A prism is made by stretching the same cross-section through a length or depth.

Key Idea

Main formula

For any prism, use $\times L$ , where $A$ is the area of the cross-section and $L$ is the length of the prism.

Example

A rectangular prism

A prism has a rectangular cross-section measuring 6 cm by 5 cm. The prism is 8 cm long. Find its volume.

Rectangular prism with a 6 cm by 5 cm rectangular cross-section stretched 8 cm long.

Find the area of the rectangular cross-section.
$\times 5 = 30\text{ cm}^2$
Multiply by the length of the prism.
$\times 8 = 240\text{ cm}^3$
The volume is 240 cm³.

Composite cross-sections

A composite shape is made by joining simpler shapes together. For an L-shaped cross-section, split it into rectangles, find each area, then add them.

Example

L-shaped prism

An L-shaped cross-section can be split into two non-overlapping rectangles: one is 5 cm by 2 cm, and the other is 3 cm by 4 cm. The prism is 10 cm deep. Work out the volume.

L-shaped prism cross-section split into two rectangles before being extended 10 cm deep.

Find the area of the first rectangle.
$cm2A_1 = 5 \times 2 = 10\text{ cm}^2$
Find the area of the second rectangle.
$cm2A_2 = 3 \times 4 = 12\text{ cm}^2$
Add the areas to get the cross-section area.
$22\text{ cm}^2$
Multiply by the depth of the prism.
$\times 10 = 220\text{ cm}^3$
The volume is 220 cm³.

Common Mistake

Adding the depth

Do not add the prism length to the cross-section area. Area is in cm² and length is in cm, so you multiply them to make cm³.

Triangular prisms

A triangular prism has a triangle as its cross-section. A right-angled triangle contains a 90° angle.

For a triangle:

\frac{1}{2}bh

where $b$ is the base and $h$ is the perpendicular height.

Example

Right-angled triangular prism

A triangular prism has a right-angled triangular cross-section with perpendicular sides 6 cm and 8 cm. The prism is 9 cm long. Find the volume.

Right-angled triangular prism with perpendicular sides 6 cm and 8 cm and length 9 cm.

Find the area of the triangular cross-section.
$\frac{1}{2} \times 6 \times 8 = 24\text{ cm}^2$
Multiply by the length of the prism.
$\times 9 = 216\text{ cm}^3$
The volume is 216 cm³.

Common Mistake

Using the hypotenuse

In a right-angled triangle, the hypotenuse is the longest side opposite the right angle. It is not needed for triangle area unless it is paired with a perpendicular height.

The hypotenuse is opposite the right angle and is not one of the perpendicular base-height pair.

Trapezium prisms

A trapezium is a four-sided shape with one pair of parallel sides. Parallel sides are always the same distance apart and never meet.

For a trapezium:

\frac{1}{2}(a+b)h

where $a$ and $b$ are the parallel sides, and $h$ is the perpendicular height between them.

Example

Trapezium cross-section

A prism has a trapezium cross-section. The parallel sides are 6 cm and 10 cm, and the perpendicular height is 5 cm. The prism is 11 cm long. Find the volume.

Trapezium prism with parallel sides 6 cm and 10 cm, perpendicular height 5 cm, and length 11 cm.

Find the area of the trapezium.
$\frac{1}{2}(6+10)\times 5 = 40\text{ cm}^2$
Multiply by the length of the prism.
$\times 11 = 440\text{ cm}^3$
The volume is 440 cm³.

Working backwards

Sometimes the question gives you the volume and asks for a missing length, often called $x$ .

Use the volume formula, then solve the equation.

Example

Finding a missing height

A triangular prism has a right-angled triangular cross-section with base 7 cm and height $x$ cm. The prism is 12 cm long. Its volume is 252 cm³. Find $x$ .

Right-angled triangular prism where the perpendicular height of the cross-section is unknown.

Write the volume equation.
$12×7×x×12=252\frac{1}{2}\times 7 \times x \times 12 = 252$
Simplify the numbers.
$42 x = 252$
Divide both sides by 42.
$\frac{252}{42} = 6$
The missing height is 6 cm.

Tip

Keep the formula visible

When working backwards, write the full volume formula first. This helps you avoid forgetting the $12\frac{1}{2}$ in triangle questions.

Units, litres and containers

For containers, volume often connects to capacity.

1 ml = 1 cm³
1 litre = 1000 cm³

Example

Filling a container with buckets

A rectangular container is 70 cm long, 50 cm wide and 40 cm high. Each bucket holds 12 litres. How many buckets are needed to fill the container?

Find the volume of the container.
$\times 50 \times 40 = 140000\text{ cm}^3$
Convert cm³ to litres.
$litres140000\text{ cm}^3 = 140\text{ litres}$
Divide by the amount in one bucket.
$140 \div 12 = 11.666\ldots$
You need 12 buckets, because 11 buckets would not completely fill the container.

Tip

Rounding in capacity questions

If you are filling a container, round up. If you are finding how many cups can be completely filled, round down.

Boxes inside a larger cuboid

For packing boxes into a van or container, check how many fit along each direction.

Example

Packing boxes

A van space is 2.4 m long, 1.5 m wide and 1.2 m high. Each box is 60 cm long, 30 cm wide and 40 cm high. How many boxes fit?

Packing identical boxes into a cuboid van space requires checking how many fit along each direction.

Convert the van dimensions to centimetres: 2.4 m is 240 cm, 1.5 m is 150 cm, and 1.2 m is 120 cm.
Find how many boxes fit along the length.
$240 \div 60 = 4$
Find how many fit along the width and height.
$150 \div 30 = 5,\qquad 120 \div 40 = 3$
Multiply the three amounts.
$\times 5 \times 3 = 60$
The van can fit 60 boxes.

Cubes and surface area

A cube is a cuboid where every edge has the same length. The surface area is the total area of all the outside faces.

A cube has 6 identical square faces.

Example

Volume from surface area

The total surface area of a cube is 294 cm². Find its volume.

Divide by 6 to find the area of one square face.
$294 \div 6 = 49\text{ cm}^2$
Find the side length of the square face.
$49=7\sqrt{49}=7$
Cube the side length to find the volume.
$7^3 = 343\text{ cm}^3$
The volume is 343 cm³.

Exam technique

In the exam

Identify the cross-section first, then find its area.
Multiply the cross-section area by the prism length or depth.
Check units carefully, especially metres to centimetres and cm³ to litres.
For missing lengths, set up an equation using the volume formula before solving.

Self review

Check yourself

Can you point to the cross-section and the length of the prism in a diagram?
Which area formula would you use for a triangle or trapezium cross-section?
When should you round up, and when should you round down, in container questions?

Volume of a Prism

What you'll learn

The basic idea

Composite cross-sections

Triangular prisms

Trapezium prisms

Working backwards

Units, litres and containers

Boxes inside a larger cuboid

Cubes and surface area

Recap questions

You've reached the end

Volume of a Prism

What you'll learn

The basic idea

Composite cross-sections

Triangular prisms

Trapezium prisms

Working backwards

Units, litres and containers

Boxes inside a larger cuboid

Cubes and surface area

Recap questions

You've reached the end

Volume of a Prism

What you'll learn

The basic idea

Composite cross-sections

Triangular prisms

Trapezium prisms

Working backwards

Units, litres and containers

Boxes inside a larger cuboid

Cubes and surface area

Recap questions

You've reached the end

Volume of a Prism Revision Guide

Volume of a Prism

What you'll learn

The basic idea

Composite cross-sections

Triangular prisms

Trapezium prisms

Working backwards

Units, litres and containers

Boxes inside a larger cuboid

Cubes and surface area

Recap questions

You've reached the end

Volume of a Prism Revision Guide