Volume questions appear on every GCSE Maths paper and can carry 3 to 5 marks each — so they are well worth mastering. Whether you are finding the volume of a simple cuboid on Foundation or working with cones and spheres on Higher, the principles are consistent: identify the shape, select the correct formula, substitute the measurements, and calculate. AQA, Edexcel and OCR all test these skills, and Higher papers often combine volume with algebra or density. This guide covers every 3D volume formula you need, with fully worked examples at both tiers and practice questions to test yourself. For a full formula reference, see our GCSE Maths formulas guide.
What Is Volume?
Volume is the amount of three-dimensional space an object occupies. It is measured in cubic units such as cm³, m³, or mm³.
Key Volume Formulas
Cube: $$V = s^3$$
Cuboid: $$V = l \times w \times h$$
Prism (any): $$V = \text{cross-sectional area} \times \text{length}$$
Cylinder: $$V = \pi r^2 h$$
Cone: $$V = \frac{1}{3}\pi r^2 h$$
Sphere: $$V = \frac{4}{3}\pi r^3$$
Hemisphere: $$V = \frac{2}{3}\pi r^3$$
Pyramid: $$V = \frac{1}{3} \times \text{base area} \times h$$
Which Formulas Are Given?
On AQA and Edexcel, the formulas for cone, sphere and pyramid are provided on the formula sheet. Cuboid, prism and cylinder formulas are not given — you must know these from memory.
Step-by-Step Method
- Identify the 3D shape. Is it a prism, cylinder, cone, sphere, or a composite solid?
- Write the formula.
- Identify the measurements. Be careful to distinguish radius from diameter and perpendicular height from slant height.
- Substitute and calculate.
- Include cubic units (cm³, m³, etc.).
- For composite solids, find the volume of each component and add or subtract.
Worked Example 1 — Foundation Level
A triangular prism has a triangular cross-section with base 6 cm and height 4 cm. The prism is 10 cm long. Find its volume.
Step 1: Find the cross-sectional area (triangle). $$A = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$$
Step 2: Volume = area × length. $$V = 12 \times 10 = \textbf{120 cm}^3$$
Cylinder Example
A cylinder has radius 3 cm and height 8 cm. Find its volume to 1 decimal place.
$$V = \pi \times 3^2 \times 8 = 72\pi = \textbf{226.2 cm}^3$$
Worked Example 2 — Higher Level
A solid is made from a cylinder topped with a hemisphere. The cylinder has radius 5 cm and height 12 cm. Find the total volume. Give your answer to 3 significant figures.
Step 1: Volume of the cylinder. $$V_{\text{cyl}} = \pi \times 5^2 \times 12 = 300\pi$$
Step 2: Volume of the hemisphere (radius 5 cm). $$V_{\text{hemi}} = \frac{2}{3}\pi \times 5^3 = \frac{250\pi}{3}$$
Step 3: Total volume. $$V = 300\pi + \frac{250\pi}{3} = \frac{900\pi + 250\pi}{3} = \frac{1150\pi}{3} = 1,204.277\ldots$$
Answer: 1,200 cm³ (3 s.f.).
Cone Example
A cone has radius 4 cm and perpendicular height 9 cm. Find its volume in terms of π.
$$V = \frac{1}{3}\pi \times 4^2 \times 9 = \frac{144\pi}{3} = \textbf{48π cm}^3$$
Common Mistakes
- Using diameter instead of radius. The formulas use r. If given the diameter, halve it first.
- Forgetting the ⅓ for cones and pyramids. A cone is one-third of the cylinder with the same base and height.
- Using slant height instead of perpendicular height. Volume formulas always require the vertical (perpendicular) height, not the slant height.
- Missing cubic units. Volume is always in cm³, m³, etc. — not cm² or cm.
- Not finding the cross-section first for prisms. The prism formula requires you to calculate the area of the cross-section before multiplying by the length.
- Rounding errors. Keep full precision until the end, then round.
Exam Tips
- State the formula — even if it is on the formula sheet, writing it in your working shows the examiner which formula you are using.
- For prisms, always identify the cross-section. It might be a triangle, trapezium, L-shape, or any other 2D shape. Find its area first.
- Composite solids are common on Higher papers. Split them into recognisable shapes, find each volume, and combine.
- Density questions often follow volume calculations: Mass = Density × Volume. Be ready to chain the methods together.
- Leave your answer in terms of π if instructed. Otherwise, use the π button on your calculator for accuracy.
- Check by estimation. A cylinder with r = 10 and h = 10 should be roughly 3,000 cm³ (π × 100 × 10 ≈ 3,142). If your answer is 300, you have probably missed a zero.
Practice Questions
Question 1 (Foundation) A cuboid measures 8 cm by 5 cm by 3 cm. Find its volume.
Question 2 (Foundation) A cylinder has diameter 10 cm and height 15 cm. Find its volume to the nearest whole number.
Question 3 (Higher) A sphere has radius 6 cm. Find its volume in terms of π.
Question 4 (Higher) A cone has a volume of 150π cm³ and a radius of 6 cm. Find the perpendicular height.
Question 5 (Higher) A hollow pipe is 2 m long with an outer radius of 5 cm and an inner radius of 3 cm. Find the volume of material in the pipe in cm³. Give your answer to 3 significant figures.
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Related Topics
- Surface Area — the total area of all faces of a 3D shape
- Area of 2D Shapes — needed for cross-sectional areas
- Compound Measures — volume connects to density calculations
- Unit Conversions — converting between cm³ and m³
- Pythagoras' Theorem — sometimes needed to find a missing height
Summary
Volume measures the three-dimensional space inside a solid and is always in cubic units. For prisms and cylinders, multiply the cross-sectional area by the length or height. For cones and pyramids, use one-third of the equivalent prism or cuboid formula. For spheres, use (4/3)πr³. Always use the perpendicular height (not the slant height), distinguish radius from diameter, and show the formula in your working. Volume questions are high-value marks on every GCSE paper, so practising the full range of shapes is time well spent.