Geometry · GCSE Maths
Volume of 3D shapes
GCSE Maths volume: prisms (cross-section × length), cuboids, cylinders, spheres (4/3πr³), units cm³ and m³, and the link to capacity in litres and millilitres.
Prism: cross-section area × length. Cylinder is a circular prism. Sphere is 4/3πr³ — cube the radius, do not confuse with 4πr².
The important bits
What you need to know
- 1
Any prism: V = cross-sectional area × length (or height). The cross-section stays the same all the way through — triangular prism, L-shaped prism, cylinder.
- 2
Cuboid: V = l × w × h. A cube is a special cuboid with l = w = h = a, so V = a³.
- 3
Cylinder: V = πr²h. The circular face is the cross-section; h is the perpendicular height between the circular ends.
- 4
Sphere: V = 4/3 πr³. Radius must be cubed. This is not the same as surface area 4πr².
- 5
Units: volume is cubic — cm³, m³, mm³. 1 m³ = 1 000 000 cm³ because 100³ = 1 000 000.
- 6
Capacity: 1 cm³ = 1 ml, so 1000 cm³ = 1 litre. A tank of 2.5 m³ holds 2 500 000 cm³ = 2 500 litres.
- 7
Compound solids: add volumes of parts, or subtract a hole (e.g. cylindrical hole drilled through a block) if the question describes removal.
- 8
Show units on every final answer. Converting m³ to litres is a common second mark on applied questions.
Quotations worth analysing
Short evidence. Real method.
“V = cross-section × length”
Find the area of the face you see at the end, then multiply by how far that shape is extruded. Cylinder, cuboid and hexagonal prism all obey this.
“V = 4/3 πr³”
The 4/3 is part of the formula, not optional. Students who write 4πr³ have confused volume with a multiple of surface area.
“1000 cm³ = 1 litre”
Applied questions often ask “how many litres?” after a volume in cm³. Divide by 1000, or convert m³ to litres via cm³.
Go deeper
Prisms: one cross-section, one length
A triangular prism has a triangular cross-section. If the triangle has base 6 cm and height 4 cm, cross-section area = ½ × 6 × 4 = 12 cm². Length of the prism 10 cm gives V = 12 × 10 = 120 cm³. An L-shaped prism: split the cross-section into two rectangles, add their areas, multiply by length. A cylinder is a prism with a circular cross-section: πr² × h. Always state “cross-section = …” as a line of working. If the question gives diameter, find r first. Units: if lengths are in cm, volume is cm³. Mixing cm and m in one calculation without converting loses marks even if the number is close.
Go deeper
Sphere versus cylinder — different powers of r
Cylinder: r appears squared (πr²h). Sphere: r appears cubed (4/3 πr³). A sphere of radius 3 cm: V = 4/3 × π × 27 = 36π cm³. Do not use 4π × 9. On calculator papers, work (4/3) × π × r³ as one chain or bracket the radius cubed first. Hemisphere volume is half the sphere: 2/3 πr³. A solid hemisphere sitting on a flat face still uses the full radius from the centre to the curved surface. Compound problems may add a hemisphere to a cylinder (ice cream cone style) — find each volume separately, then add.
Go deeper
Capacity and real-world units
Volume in cm³ converts directly to millilitres: 500 cm³ = 500 ml. Litres: divide cm³ by 1000. A swimming pool 4 m × 10 m × 1.5 m: V = 60 m³ = 60 × 1 000 000 cm³ = 60 000 000 cm³ = 60 000 litres. Examiners may ask how many 2-litre bottles fill a tank — divide total litres by 2. When a question gives dimensions in metres but asks for litres, either convert to cm³ early or use 1 m³ = 1000 litres directly. Write the conversion line explicitly: “60 m³ × 1000 = 60 000 litres.”
See the idea in action
A cylinder has radius 3 cm and height 10 cm. Find its volume, giving your answer in terms of π. Step 1: Cylinder is a prism with circular cross-section. V = πr²h. Step 2: r = 3 cm, h = 10 cm. Step 3: V = π × 3² × 10 = π × 9 × 10. Step 4: V = 90π cm³. Check: r² = 9, times height 10 gives 90; units are cm³.
Exam technique
Turn knowledge into marks
Write “cross-section area = …” then “× length = …” for prisms. For spheres, write 4/3 × π × r³ with r substituted before you press equals.
Common mistakes
Do not give these marks away
- 01
Using 4πr² or 2πr²h for sphere volume instead of 4/3 πr³.
- 02
Forgetting to square the radius in πr²h, or using diameter in place of radius.
- 03
Confusing cm³ with litres (off by a factor of 1000) or mixing m and cm without converting before multiplying three lengths.
A cuboid is 4 cm × 5 cm × 6 cm. What is its volume?
A120 cm³
B60 cm³
C15 cm³
D240 cm³
Show the answer
120 cm³. V = l × w × h = 4 × 5 × 6 = 120 cm³. 60 cm³ might come from adding instead of multiplying. 15 cm³ is half of a face area, not volume.
Quick questions
If this is the bit you searched
What is the difference between a prism and a cylinder?
A cylinder is a prism whose cross-section is a circle. Both use volume = cross-sectional area × length.
How do I convert m³ to litres?
1 m³ = 1 000 000 cm³ = 1000 litres. Multiply m³ by 1000 to get litres directly.
When do I use 4/3 πr³?
For the volume of a full sphere. For a hemisphere, use half of that: 2/3 πr³.
Can I add volumes of different shapes?
Yes, if the solid is made of separate parts (e.g. cylinder plus hemisphere). Use the same units throughout before adding.