Volume of a Sphere

GCSE Geometry volume sphere
Practice this formula All formulas
\( V=\tfrac{4}{3}\pi r^3 \)

Statement

The volume of a sphere is given by:

\[ V = \tfrac{4}{3}\pi r^3 \]

where \(r\) is the radius of the sphere.

Why it’s true

  • A sphere is the 3D set of points equidistant from a center.
  • Its volume can be derived by integration in calculus or by geometric comparison with a cylinder and cone (Archimedes’ theorem).
  • The factor \(\tfrac{4}{3}\) ensures the formula fits these derivations.

Recipe (how to use it)

  1. Find the radius (remember radius = half of diameter).
  2. Cube it (\(r^3\)).
  3. Multiply by \(\pi\).
  4. Multiply by \(\tfrac{4}{3}\).
  5. Answer in cubic units.

Spotting it

Look for perfect ball shapes — e.g. footballs, oranges, marbles.

Common pairings

  • Often asked alongside surface area of a sphere (\(4\pi r^2\)).
  • May be used in density and mass problems.

Mini examples

  1. r=3: \(V=\tfrac{4}{3}\pi(27)=36\pi\).
  2. r=5: \(V=\tfrac{4}{3}\pi(125)=500/3\pi\).

Pitfalls

  • Using diameter instead of radius.
  • Forgetting to cube the radius.
  • Mixing with surface area formula.

Exam strategy

  • Always write the formula first.
  • Check whether diameter or radius is given.
  • Leave answers in terms of \(\pi\) unless decimals are required.

Summary

The volume of a sphere is \(\tfrac{4}{3}\pi r^3\). Cube the radius, multiply by \(\pi\), then scale by \(\tfrac{4}{3}\).