Find the volume of the cube.
Volume of cube = side³.
4³ = 64 cm³.
Volume
Volume measures the space inside a 3D shape. It links closely to surface area and unit conversions, especially when working with cubic units.
Overview
Volume tells you how much space is inside a 3D shape.
It is measured in cubic units, such as \( \text{cm}^3 \), \( \text{m}^3 \) or \( \text{mm}^3 \).
\( \text{Volume} = \text{area of cross-section} \times \text{length} \)
In GCSE Maths, many volume questions are about choosing the correct formula, substituting values carefully, and remembering the correct units.
What you should understand after this topic
- Understand what volume means
- Choose the correct volume formula for each shape
- Calculate volume of prisms and cylinders
- Use formulas for cones, pyramids and spheres
- Use cubic units correctly
Key Definitions
Volume
The amount of space inside a 3D shape.
Cubic Units
Units used for volume, such as \( \text{cm}^3 \) or \( \text{m}^3 \).
Prism
A 3D shape with the same cross-section all the way through.
Cross-section
The shape made by slicing through a solid.
Radius
The distance from the centre of a circle to its edge.
Height
The vertical distance from base to top.
Base Area
The area of the bottom face used in some volume formulas.
Capacity
A measure of how much a container can hold, often in litres or millilitres.
Key Rules
Volume measures space
Volume is the amount of space inside a three-dimensional object.
Use the correct formula
Each 3D shape has its own volume formula, so identify the shape first.
Multiply area by height for prisms
Volume of a prism = area of cross-section × length.
Always include cubic units
Write answers in units such as cm³, m³, or mm³.
Key Formulas
Cuboid
\( V = l \times w \times h \)
Cube
\( V = a^3 \)
Prism
\( V = \text{Area of cross-section} \times \text{length} \)
Cylinder
\( V = \pi r^2 h \)
Cone
\( V = \frac{1}{3} \pi r^2 h \)
Sphere
\( V = \frac{4}{3} \pi r^3 \)
Pyramid
\( V = \frac{1}{3} \times \text{Base Area} \times h \)
How to Solve
Step 1: Understand volume
Volume measures the space inside a 3D shape.
Volume is measured in cubic units.
Examples: \(\text{cm}^3\), \(\text{m}^3\).
Step 2: Identify the shape
Cuboid
Multiply 3 dimensions
Prism
Cross-section × length
Cylinder
Circle × height
Cone / Pyramid
Use \(\frac{1}{3}\)
Step 3: Key formulas
Cuboid: \( V = lwh \)
Prism: \( V = \text{area} \times \text{length} \)
Cylinder: \( V = \pi r^2h \)
Cone: \( V = \frac{1}{3}\pi r^2h \)
Pyramid: \( V = \frac{1}{3}\times \text{base area} \times h \)
Sphere: \( V = \frac{4}{3}\pi r^3 \)
Exam tip: Always write the formula first.
Step 4: Radius and height
Check radius vs diameter.
Diameter → divide by 2.
Use perpendicular height.
Step 5: Prisms
Same cross-section throughout.
\( V = \text{area of cross-section} \times \text{length} \)
Includes cuboids, cylinders and triangular prisms.
Step 7: Exam method
- Identify the shape.
- Choose formula.
- Check measurements.
- Substitute values.
- Calculate.
- Write cubic units.
Example Questions
Edexcel
Exam-style questions focusing on volume of cuboids and cubes.
Edexcel
A cuboid has dimensions 6 cm, 4 cm and 3 cm.
Find the volume of the cuboid.
Edexcel
A cube has side length 5 cm.
Find the volume of the cube.
AQA
Exam-style questions focusing on prisms and cylinders.
AQA
A prism has cross-section area 15 cm² and length 11 cm.
Find the volume of the prism.
AQA
A cylinder has radius 3 cm and height 8 cm.
Find the volume of the cylinder in terms of π.
OCR
Exam-style questions focusing on cones, spheres and formula-based solids.
OCR
A cone has radius 4 cm and height 9 cm.
Find the volume of the cone in terms of π.
OCR
A sphere has diameter 10 cm.
Find the volume of the sphere in terms of π.
OCR
A solid prism has cross-section area A and length l.
Write the formula for the volume of a prism.
Exam Checklist
Step 1
Identify the 3D shape correctly.
Step 2
Choose the correct volume formula.
Step 3
Substitute values carefully, especially radius and height.
Step 4
Write the answer in cubic units.
Most common exam mistakes
Radius mistake
Using the diameter instead of halving it first.
Formula mistake
Using the cylinder formula for a cone, or forgetting the \( \frac{1}{3} \).
Units mistake
Writing \( \text{cm} \) or \( \text{cm}^2 \) instead of \( \text{cm}^3 \).
Shape mistake
Confusing surface area and volume.
Common Mistakes
These are common mistakes students make when calculating volume in GCSE Maths.
Using area units instead of cubic units
Incorrect
A student gives the answer in \(\text{cm}^2\) instead of \(\text{cm}^3\).
Correct
Volume is measured in cubic units, such as \(\text{cm}^3\) or \(\text{m}^3\). Always include the correct units.
Using diameter instead of radius
Incorrect
A student uses the full width in place of the radius.
Correct
If a diameter is given, divide by 2 to find the radius before using formulas involving \(\pi r^2\).
Forgetting the \(\frac{1}{3}\) factor
Incorrect
A student calculates volume of a cone or pyramid without dividing by 3.
Correct
The volume of cones and pyramids is \(\frac{1}{3} \times \text{base area} \times \text{height}\).
Confusing volume with surface area
Incorrect
A student uses surface area formulas instead of volume.
Correct
Volume measures the space inside a shape, while surface area measures the outside covering. Use the correct formula.
Incorrect rounding with \(\pi\)
Incorrect
A student rounds too early or inconsistently.
Correct
Keep \(\pi\) in your calculations until the final step, then round to the required accuracy.
Try It Yourself
Practise calculating the volume of 3D shapes.
Foundation
Foundation Practice
Calculate volume of cubes, cuboids and simple prisms.
Find the volume of a cuboid with length 6 cm, width 4 cm and height 3 cm.
Volume = length × width × height.
6 × 4 × 3 = 72 cm³.
Find the volume of the cuboid.
Multiply all three dimensions.
10 × 5 × 4 = 200 cm³.
A cube has side length 7 cm. Find its volume.
Volume = side³.
7³ = 343 cm³.
Which formula is correct for volume of a cuboid?
Volume is 3D.
Volume = length × width × height.
Find the volume of a prism with cross-sectional area 12 cm² and length 8 cm.
Volume = area × length.
12 × 8 = 96 cm³.
Find the volume of the triangular prism.
Area of triangle × length.
Area = 1/2 × 6 × 4 = 12, then 12 × 10 = 120 cm³.
A cuboid has volume 150 cm³ and base area 30 cm². Find its height.
Height = volume ÷ base area.
150 ÷ 30 = 5 cm.
What are the units for volume?
Volume is 3D.
Volume is measured in cubic units (cm³).
Find the volume of a cuboid with dimensions 8 cm, 3 cm and 2 cm.
Multiply all dimensions.
8 × 3 × 2 = 48 cm³.
Higher
Higher Practice
Calculate volumes of cylinders, cones, spheres and compound solids.
Find the volume of the cylinder in terms of π.
Volume = πr²h.
π × 7² × 10 = 490π cm³.
Find the volume of a cylinder with radius 3 cm and height 8 cm in terms of π.
Use πr²h.
π × 3² × 8 = 72π cm³.
Find the volume of the cone.
Volume = 1/3 πr²h.
1/3 × π × 6² × 9 = 108π cm³.
Find the volume of a sphere with radius 3 cm in terms of π.
Volume = 4/3 πr³.
4/3 × π × 27 = 36π cm³.
A cube has volume 125 cm³. Find the side length.
Cube root.
Side = ∛125 = 5 cm.
A cylinder has volume 200π cm³ and height 10 cm. Find its radius.
Rearrange πr²h = volume.
πr² × 10 = 200π → r² = 20 → r = √20.
Which formula gives the volume of a sphere?
Sphere formula.
Volume of a sphere = 4/3 πr³.
A cone has volume 100 cm³ and height 6 cm. Find its base area.
Rearrange V = 1/3 × base area × height.
100 = 1/3 × A × 6 → A = 50 cm².
A sphere has radius doubled. What happens to its volume?
Volume depends on r³.
Doubling radius multiplies volume by 2³ = 8.
Find the volume of a cuboid with dimensions 2 m, 3 m and 4 m.
Multiply dimensions.
2 × 3 × 4 = 24 m³.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is volume?
The space inside a 3D shape.
What are common units?
Cubic units like cm³.
What is a common mistake?
Forgetting to cube units.