Volume of a Sphere Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Volume of a Sphere.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The amount of three-dimensional space inside a sphere, given by \frac{4}{3}\pi r^3.

Imagine filling a sphere with water, then pouring all that water into a cylinder that has the same radius and a height equal to the sphere's diameter (2r). The sphere fills exactly two-thirds of the cylinder. Archimedes was so proud of discovering this relationship that he had it carved on his tombstone.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The sphere volume formula has r^3 because it measures 3D space, and the \frac{4}{3} factor arises from the sphere's symmetry.

Common stuck point: The radius is cubed (r^3), not squared. Cubing makes volume grow very fast: double the radius, 8\times the volume.

Worked Examples

Example 1

easy
A basketball has a radius of 12 cm. Find its volume. Leave your answer in terms of \pi.

Solution

  1. 1
    Step 1: Write the formula: V = \frac{4}{3}\pi r^3.
  2. 2
    Step 2: Substitute r = 12: V = \frac{4}{3}\pi (12)^3 = \frac{4}{3}\pi \times 1728.
  3. 3
    Step 3: Simplify: \frac{4}{3} \times 1728 = 4 \times 576 = 2304. So V = 2304\pi cmยณ.

Answer

V = 2304\pi cmยณ.
The sphere formula V = \frac{4}{3}\pi r^3 involves cubing the radius, so even small changes in radius have a large effect on volume. Be careful to cube the radius (not the diameter) and then multiply by \frac{4}{3}\pi.

Example 2

medium
A sphere has a volume of \frac{500\pi}{3} cmยณ. Find its radius.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A sphere has a diameter of 10 cm. Find its volume. Leave your answer in terms of \pi.

Example 2

hard
If the radius of a sphere is doubled, by what factor does its volume increase? Prove your answer algebraically.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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