Sheet № 46 · Foundation + Higher · AQA · Edexcel · OCR
Volume of 3D Shapes –
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, a
§Key definitions
Step 1:
Find the cross-sectional area (triangle).
Step 2:
Volume = area × length.
Step 3:
Total volume.
Answer: 1,200 cm³
(3 s.f.).
§Formulas to memorise
V = s^3
V = l \times w \times h
V = \text{cross-sectional area} \times \text{length}
V = \pi r^2 h
V = \frac{1}{3}\pi r^2 h
V = \frac{4}{3}\pi r^3
V = \frac{2}{3}\pi r^3
V = \frac{1}{3} \times \text{base area} \times h
A = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2
V = 12 \times 10 = \textbf{120 cm}^3
V = \pi \times 3^2 \times 8 = 72\pi = \textbf{226.2 cm}^3
V_{\text{cyl}} = \pi \times 5^2 \times 12 = 300\pi
Worked example
See example below.
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.
⚠ 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.
✦ 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.