Spheres & cones – GCSE Maths

Foundation and Higher · Grade 5 · Calculator

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Sphere: $V=\frac{4}{3}\pi r^3$, surface area $4\pi r^2$. Cone: $V=\frac{1}{3}\pi r^{2}h$, curved surface $\pi rl$ where $l$ is the slant height. A hemisphere is half a sphere, plus a circle if the flat face is included. Write the formula before you substitute — it is a mark.

Slant height, perpendicular height and radius of a cone form a right-angled triangle, so Pythagoras links $l$, $h$ and $r$. Leave in terms of $\pi$ when asked. A composite ice-cream (cone plus hemisphere) is two volumes added, watching whether the hemisphere's flat face is internal (not in the surface area).

Sphere volume $\frac{4}{3}\pi r^3$, surface $4\pi r^2$. Cone volume $\frac{1}{3}\pi r^{2}h$ with $h$ the perpendicular height, not the slant. Leave $\pi$ on non-calc; 3 s.f. on calc.

Objectives

• Find the volume of a sphere in terms of π.
• Find a sphere surface area, and a hemisphere volume.
• Find the volume of a cone in terms of π.

Key points

• Substitute into $V=\frac{4}{3}\pi r^3$ and $A=4\pi r^2$ for a sphere.
• Halve the sphere volume for a hemisphere (and be careful with its surface area).
• Substitute into $V=\frac{1}{3}\pi r^{2}h$ for a cone, using the perpendicular height.
• Find the slant height $l$ with Pythagoras and use curved surface area $=\pi rl$.

Worked examples

Questions and answers

GCSE Maths spheres & cones questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Using r^2 instead of r^3 for sphere volume

   Sphere volume requires cubing the radius: (4/3)πr^3

2. Forgetting to divide by 2 for a hemisphere

   A hemisphere is half a sphere; always divide the full volume by 2

3. Forgetting the 1/3 factor in cone volume

   A cone is one third of a cylinder with the same dimensions

Exam tips

• A calculator is useful here. Show the calculation you typed, not only the final line.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Write the formula with the numbers substituted before you reach for the calculator — that line is the method mark.
• Keep answers as multiples of $\pi$ in your working; it avoids rounding errors and is sometimes the form the question wants.

Printable worksheet

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More on Spheres & cones

• Spheres & cones worked examples and method
  The method in order, key facts and the full lesson explanation
• Spheres & cones practice questions
  A second set of questions with answers, plus quick checks

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Last updated 9 November 2026.