Volume GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about volume

A cube with edges 1 cm occupies one cubic centimetre, written 1 cm³. Imagine filling a box with these cubes without gaps. If each layer contains 12 cubes and there are two layers, the box contains 24 cubes: its volume is 24 cm³.

See the idea first

Base area counts one layer; height counts the layers

For a cuboid, length × width gives the area of the rectangular base. Multiplying by height fills the whole box. For any right prism, the same cross-section continues unchanged along its length, so volume = cross-sectional area × length.

The base covers 4 × 3 = 12 cm². Two layers, each 1 cm high, fill 24 cm³. This perspective sketch is not a measuring diagram.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Count faces, edges and vertices

What the problem asks: How many faces, edges and vertices has a triangular prism?

How to solve it: A face is a flat surface; an edge is where faces meet; a vertex is a corner. Its two triangular ends and three rectangular side faces give 5 faces. The ends have 6 edges in total and 3 edges join corresponding vertices, giving 9 edges and 6 vertices. A cylinder instead has two circular flat faces and one curved surface.

Find a prism's volume

What the problem asks: A triangular prism has cross-sectional area 15 cm² and length 8 cm. Find volume.

How to solve it: The same 15 cm² section continues for 8 cm, so volume = 15 × 8 = 120 cm³.

Find a cylinder's volume

What the problem asks: A cylinder has radius 3 cm and height 7 cm. Find volume.

How to solve it: Its constant circular cross-section has area πr² = 9π cm². Multiply by height: 63π cm³.

Find a tapered solid's volume

What the problem asks: A pyramid has base area 24 cm² and perpendicular height 9 cm. Find volume.

How to solve it: A pyramid has one third the volume of the prism with the same base and perpendicular height. Use ⅓ × 24 × 9 = 72 cm³. The section is not constant, so the prism rule alone would overcount.

Find an unknown dimension

What the problem asks: A cuboid has volume 180 cm³ and base area 30 cm². Find height.

How to solve it: Each centimetre of height contributes 30 cm³. Divide 180 by 30 to get 6 cm.

A reliable routine

Calculate a prism's volume

Area × length applies to right prisms and cylinders because every perpendicular cross-section has the same area. It does not apply unchanged to a cone, pyramid or sphere.

Identify the repeated cross-section and its area, in square units.
Find the perpendicular length through which that section continues.
Convert all dimensions to compatible units and multiply.
For a compound solid, add separate non-overlapping volumes or subtract a cavity.

Check: A cone or pyramid uses perpendicular height, not the slant height. Capacity conversions: 1 cm³ = 1 ml and 1000 cm³ = 1 litre.

Fully worked

Volume GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Cuboid

2 marks

Question

Find the volume of a cuboid measuring 8 cm by 5 cm by 3 cm.

Base area is $8(5)=40$ cm².

V=40(3)=120\text{ cm}^3

Tip: three lengths produce cubic units.

Example 2

Triangular prism

3 marks

Question

A right prism has a triangular cross-section with base 6 cm and perpendicular height 5 cm. The prism is 9 cm long. Find volume.

First calculate the triangle area.

A=\frac12(6)(5)=15\text{ cm}^2

V=15(9)=135\text{ cm}^3

The 9 cm is the prism length, not the triangle height.

Example 3

Cylinder

3 marks

Question

A cylinder has diameter 8 cm and height 10 cm. Find volume in terms of π.

Radius is half the diameter: r = 4 cm.

V=\pi r^2h

=\pi(4)^2(10)

=160\pi\text{ cm}^3

Example 4

Pyramid

3 marks

Question

A square-based pyramid has base side 6 cm and perpendicular height 10 cm. Find its volume using V = one third × base area × perpendicular height.

A_{\rm base}=6^2=36\text{ cm}^2

V=\frac13(36)(10)=120\text{ cm}^3

Use the perpendicular height from the apex to the base plane.

Example 5

Cone

3 marks

Question

A cone has radius 3 cm and perpendicular height 8 cm. Use $V=\frac13\pi r^2h$ and leave your answer in terms of π.

V=\frac13\pi(3)^2(8)

=\frac13(72\pi)

=24\pi\text{ cm}^3

It is one third of the matching cylinder's volume.

Example 6

Sphere

3 marks

Question

A sphere has radius 6 cm. Use $V=\frac43\pi r^3$ to find its volume in terms of π.

V=\frac43\pi(6)^3

=\frac43\pi(216)

=288\pi\text{ cm}^3

Tip: the radius is cubed here, not squared.

11 original questions · total 29 marks

Volume GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 34 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Count a pyramid’s features

3 marks

A square-based pyramid has four triangular side faces. State its total faces, edges and vertices.

Show worked answer

Faces: 4 triangular sides + 1 square base = 5. Edges: 4 around the base + 4 rising to the apex = 8. Vertices: 4 base corners + 1 apex = 5. Do not count only the visible edges of a sketch.

2

Cube

2 marks

Find the volume of a cube of side 4 cm.

Show worked answer

V=4^3=4\times4\times4=64\text{ cm}^3

3

Cuboid

2 marks

A cuboid is 7 cm by 6 cm by 2 cm. Find volume.

Show worked answer

V=7(6)(2)=84\text{ cm}^3

4

Missing length

2 marks

A prism has volume 216 cm³ and cross-sectional area 18 cm². Find length.

Show worked answer

l=216/18=12\text{ cm}

Divide the volume into layers of the stated area.

5

Trapezoidal prism

3 marks

A prism's cross-section is a trapezium with parallel sides 4 cm and 8 cm and perpendicular height 3 cm. Its length is 10 cm. Find volume.

Show worked answer

A=\frac12(4+8)(3)=18\text{ cm}^2

V=18(10)=180\text{ cm}^3

6

Cylinder

2 marks

A cylinder has radius 5 cm and height 4 cm. Find volume in terms of π.

Show worked answer

V=\pi(5)^2(4)=100\pi\text{ cm}^3

7

Capacity

2 marks

A tank has internal dimensions 40 cm by 25 cm by 30 cm. Find its capacity in litres.

Show worked answer

V=40(25)(30)=30000\text{ cm}^3

30000/1000=30\text{ litres}

Internal dimensions measure usable space.

8

Cone

3 marks

A cone has radius 4 cm and perpendicular height 9 cm. Use $V=\frac13\pi r^2h$ .

Show worked answer

V=\frac13\pi(4)^2(9)

=48\pi\text{ cm}^3

9

Hemisphere

3 marks

A hemisphere has radius 3 cm. Use sphere volume $V=\frac43\pi r^3$ to find the hemisphere's volume.

Show worked answer

A hemisphere is half a sphere.

V=\frac12\cdot\frac43\pi(3)^3

=18\pi\text{ cm}^3

10

A hollow cylinder

Harder4 marks

A tube has outer radius 5 cm, inner radius 3 cm and length 12 cm. Find the volume of material in terms of π.

Show worked answer

Subtract the empty inner cylinder.

V=\pi(5)^2(12)-\pi(3)^2(12)

=300\pi-108\pi

=192\pi\text{ cm}^3

11

A frustum with given cone volumes

3 marks

A small cone of volume $18\pi$ cm³ is cut from a larger cone of volume $150\pi$ cm³ to leave a frustum. Find the remaining volume.

Show worked answer

The removed and remaining parts fill the original cone without overlap.

V=150\pi-18\pi=132\pi\text{ cm}^3

A frustum is the truncated cone left after the top is removed.

Examiner-style feedback

Common volume mistakes

Using surface area

Volume fills a solid; surface area wraps it. Use cubic units for volume.

Using diameter as radius

Halve a circle's diameter before using πr² or a sphere formula.

Multiplying all labels

Choose dimensions for the actual cross-section and perpendicular length; not every labelled side belongs in the formula.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Identify the solid.
Find a repeated section when it exists.
Use a valid formula for tapering or curved solids.
Keep cubic units and convert capacity carefully.

Quick answers

Volume FAQ

Is every solid a prism?

No. A prism has the same perpendicular cross-section along its length. A cone or pyramid tapers.

Does doubling a cube's edge double its volume?

No. All three dimensions double, so its volume becomes eight times as large.

Build connected skills

What to revise next

Wrapping instead of filling

Surface area

Find the area of exposed surfaces.

Revise surface area

Use a compound measure

Density, mass and volume

Relate occupied volume to mass.

Revise density, mass and volume

Content standards

Curriculum and rights review

G12, G16–G17 volumes of prisms and common solids across tiers, with formulas supplied for pyramids, cones and spheres in the relevant questions. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content