KS3 · Maths

Volume of prisms (including cuboids)

Cut a prism anywhere along its length and the slice is always the same shape. It's one flat shape, stacked up — so one rule finds every prism's volume.

Maths · Volume of prisms

Stretch the prism — watch the volume keep pace

Share of the 6 cm prism: 6/12. Length 3 cm. Area × length 4 cm² × 3 cm. Volume 12 cm³

Length3 cmArea × length4 cm² × 3 cmVolume12 cm³

This prism's cross-section is fixed at 4 cm². Each box on the bar is half a centimetre of length. Drag the length: every extra centimetre stacks one more 4 cm² layer, so the volume grows by 4 cm³ — and half a centimetre adds half a layer, 2 cm³.

Layers and slices

Does the way you count change the volume?

Where do layers come from? Picture a cuboid built from cubes: 3 cubes wide, 5 cubes long and 2 cubes high. Aisha counts it as 2 layers of 15. Ben counts it as 5 slices of 6.

Which is closest to what you think right now?

They'll get different answers — cutting it a different way changes how much space it takes up.If you filled the same cuboid with smaller cubes, its volume would go up, because more cubes would fit.Rebuild the same cubes into a different cuboid and the volume will change, because the shape has changed.Layers or slices, it's 30 cubes either way — the cuboid takes up the same space however you count it.

How sure are you?

I'm sureI think soI'm guessing

They'll get different answers — cutting it a different way changes how much space it takes up.

This is a common misconception

Try both counts: 2 layers of 15 is 30, and 5 slices of 6 is 30. You could even do 3 slices of 10 — still 30. Layers and slices are just ways of organising the count. The cuboid itself hasn't changed, so the space it takes up can't change either.

If you filled the same cuboid with smaller cubes, its volume would go up, because more cubes would fit.

This is a common misconception

You'd count more cubes — but each one is smaller, so they fill exactly the same space. A box that holds 3 layers of 15 one-centimetre cubes has a volume of 45 cm³, and it stays 45 cm³ whatever size cubes you fill it with. The number of cubes depends on the cube size; the volume doesn't.

Rebuild the same cubes into a different cuboid and the volume will change, because the shape has changed.

This is a common misconception

The same 24 cubes can be put together to make several different cuboids — long and thin, short and wide. Each one is made of exactly those 24 cubes, so each takes up exactly the same amount of space. The shape changes; the volume doesn't.

Layers or slices, it's 30 cubes either way — the cuboid takes up the same space however you count it.

This matches the evidence

Exactly. 2 × 15, 3 × 10 and 5 × 6 all give 30. And that's the seed of the whole lesson: a layer is one copy of the cross-section, so counting layers is really multiplying the area of the cross-section by the length.

Counting in layers or in slices only organises the count. The space inside the cuboid — its volume — is fixed.

The method, step by step

Problem

A prism has a right-angled triangle as its cross-section. The two sides that meet at the right angle are 3 cm and 4 cm. The prism is 8 cm long. Find its volume.

Spot the mistake

Where does this answer go wrong?

A swimming pool is 50 m long and 18 m wide. It is 1.2 m deep at the shallow end and 1.8 m deep at the deep end, and the bottom slopes evenly between them. Find the volume of the pool.

A student's answer — which line goes wrong?

The measurements are: 50 m long, 18 m wide, and depths of 1.2 m and 1.8 m.
Volume is found by multiplying the lengths together, so multiply all four: 50 × 18 × 1.2 × 1.8
= 1944
So the volume of the pool is 1944 m³.

Predict, then check

Don't calculate yet — decide first.

A cuboid measures 50 mm by 8 cm by 3 cm. You want its volume in cm³. What's your first move?

Working backwards

Now you supply the missing steps

A triangular prism has a volume of 84 cm³. Its cross-section is a triangle with base x cm and perpendicular height 4 cm. The prism is 7 cm long. Find x.

Start from the rule: volume = area of cross-section × length.When the unknown is the whole area or the length, it's one division. A prism of volume 344 m³ and length 8 m: 344 = 8a, so a = 43 m². A prism with cross-section 52 cm² and volume 379.6 cm³: height = 379.6 ÷ 52 = 7.3 cm. Here the unknown is inside the triangle, so write the area in terms of x first.
missing step

Which line is step 2?

Area of the triangle = ½ × x × 4 = 2xArea of the triangle = x × 4 = 4xArea of the triangle = x × 4 × 7 = 28x

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

Volume is the amount of space a closed 3D shape takes up, measured in cubic units: mm³, cm³, m³, km³.
A prism has the same cross-section all the way along its length.
Volume of any prism = area of cross-section × length.
Make all the units match before you multiply.
To find a missing measurement, substitute into the formula and use inverse operations.

The big picture

Volume is the amount of space a closed 3D shape takes up, measured in cubic units such as cm³ or m³. A prism has the same cross-section all the way along its length, so its volume is the area of that cross-section multiplied by the length. The method works for any cross-section — rectangle, triangle, trapezium, pentagon — and for lengths that aren't whole numbers. Get the units matching before you multiply, and to find a missing length or area, substitute into the formula and use inverse operations.

Key points

1A cross-section is the flat face you get by cutting straight through a 3D shape. In a prism it's the same shape and size wherever you cut along the length.

2Each extra unit of length adds one more layer the shape of the cross-section — so volume = area of cross-section × length, even when the length isn't a whole number.

3The 'length' is the perpendicular distance between the two identical end faces. Depending on how the prism sits, it might be called the height.

4For a cuboid, any face can be the cross-section — you get the same volume every time.

5Never multiply every number on the diagram: find the cross-section's area first, then multiply by the length.

6Dividing a volume by a length takes away one dimension, so a missing area comes out in square units (m³ ÷ m = m²).

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

Carton A is a cuboid with a base 4 cm by 5 cm and an unknown height x cm. It holds exactly the same amount of liquid as carton B, a cuboid measuring 7 cm by 5 cm by 6.4 cm. Find x.

⚠ Watch out

Multiplying every number on the diagram. A trapezium or triangle cross-section has measurements that belong to the flat shape — combine those into an area first, and only then multiply by the length. If you've multiplied four lengths together, it can't be a volume.

🧠

Memory hook

Slice it, find it, stack it: slice the prism to see its cross-section, find that shape's area, then stack it along the length.

✓

Check yourself

Without looking back: a prism has a volume of 150 m³ and a length of 6 m. What is the area of its cross-section — and why is the answer in m², not m³?

Flashcards

(14)

What is volume?

The amount of space a closed 3D shape takes up.

Why is volume measured in cubic units?

Finding any volume multiplies three perpendicular lengths together somewhere in the calculation — for example cm × cm × cm = cm³.

What makes a solid a prism?

Its cross-section is the same shape and size all the way along its length.

What is a cross-section?

The 2D face you get by cutting straight through a 3D object.

The volume formula for any prism

Volume = area of cross-section × length.

In a prism, what does 'length' mean?

The perpendicular distance between the two identical end faces. Some diagrams call it the height.

Which face of a cuboid is its cross-section?

Any of them. Multiplication is commutative, so every choice gives the same volume.

A cuboid is 3 cubes wide, 5 long and 2 high. Count it in layers, then in slices.

2 layers of 15 = 30. Or 3 slices of 10, or 5 slices of 6 — always 30 cubes.

Does filling a box with smaller cubes change its volume?

No. You count more cubes, but the space inside the box stays exactly the same.

The measurements are in mixed units. What do you do first?

Make them all the same unit before multiplying. Converting the one odd measurement is usually quickest — e.g. 50 mm → 5 cm.

Area of a trapezium cross-section

Half the sum of the parallel sides × the perpendicular distance between them.

How do you find a missing length when you know the volume?

Put the numbers you know into the volume formula, then undo the multiplication — usually by dividing.

Why does a missing cross-section area come out in square units?

Dividing a volume by a length takes away one dimension: m³ ÷ m = m².

Two containers hold the same amount of liquid. What can you write down?

Their volumes are equal — so set volume of one = volume of the other and solve.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Does the way you count change the volume?

Where does this answer go wrong?

Now you supply the missing steps

More KS3 Maths topics