Volume of a Sphere

\( V=\tfrac{4}{3}\pi r^3 \)
Geometry GCSE
Question 5 of 20
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Sphere radius 9 cm. Find volume.

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Cube radius first.

Explanation

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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 \(\tfrac{4}{3}\).

Worked examples

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  1. Find the volume of a sphere with radius 3 cm.
    1. \( r^3=27 \)
    2. \( V=4/3 π×27=36π \)
    Answer: 36π cm³
  2. Find the volume of a sphere with radius 5 cm.
    1. \( r^3=125 \)
    2. \( V=4/3 π×125=500/3 π \)
    Answer: 500/3 π cm³
  3. Sphere with diameter 10 cm. Volume?
    1. \( r=5 \)
    2. \( r^3=125 \)
    3. \( V=500/3 π \)
    Answer: 500/3 π cm³
  4. Sphere with radius 7 cm. Volume?
    1. \( r^3=343 \)
    2. \( V=4/3 π×343=1372/3 π \)
    Answer: 1372/3 π cm³
  5. Sphere with radius 1 cm. Volume?
    1. \( r^3=1 \)
    2. \( V=4/3 π \)
    Answer: 4/3 π cm³
  6. Find volume of sphere radius 12 cm.
    1. \( r^3=1728 \)
    2. \( V=4/3 π×1728=2304π \)
    Answer: 2304π cm³
  7. Sphere radius 10 cm. Volume?
    1. \( r^3=1000 \)
    2. \( V=4/3 π×1000=4000/3 π \)
    Answer: 4000/3 π cm³
  8. Sphere diameter 18 cm. Volume?
    1. \( r=9 \)
    2. \( r^3=729 \)
    3. \( V=4/3 π×729=972π \)
    Answer: 972π cm³
  9. Sphere radius 2.5 cm. Volume?
    1. \( r^3=15.625 \)
    2. \( V=4/3 π×15.625=20.833...π \)
    Answer: 20.833π cm³
  10. General: radius r. Volume?
    1. \( V=4/3 πr^3 \)
    Answer: \( 4/3 πr^3 \)