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 43πr3\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 (2r2r). 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: A sphere is a round solid whose size is controlled by radius in every direction.

Common stuck point: The procedure for volume of a sphere is the easy part; the trap is using diameter as radius. Asking "Is the solid round in every direction with points equally far from a center?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the solid round in every direction with points equally far from a center?

Worked Examples

Example 1

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

Answer

V=2304πV = 2304\pi cm³.

First step

1
Step 1: Write the formula: V=43πr3V = \frac{4}{3}\pi r^3.

Full solution

  1. 2
    Step 2: Substitute r=12r = 12: V=43π(12)3=43π×1728V = \frac{4}{3}\pi (12)^3 = \frac{4}{3}\pi \times 1728.
  2. 3
    Step 3: Simplify: 43×1728=4×576=2304\frac{4}{3} \times 1728 = 4 \times 576 = 2304. So V=2304πV = 2304\pi cm³.
The sphere formula V=43πr3V = \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 43π\frac{4}{3}\pi.

Example 2

medium
A sphere has a volume of 500π3\frac{500\pi}{3} cm³. Find its radius.

Example 3

medium
A solid sphere has volume 972π972\pi. Find its radius.

Example 4

medium
A spherical ball just fits inside a cube of side 1010. Find the volume of the empty space inside the cube, in terms of π\pi.

Example 5

medium
A balloon's volume doubles. By what factor does its radius grow? Give answer to 3 decimal places.

Example 6

hard
A spherical ball of radius 33 is dropped into a cylinder of radius 33 partly full of water. By how much does the water level rise?

Example 7

hard
A sphere is inscribed in a cylinder so that the cylinder's height equals the sphere's diameter. Show the ratio of sphere volume to cylinder volume is 2:32:3.

Example 8

challenge
A spherical cap of height hh is cut from a sphere of radius RR. Use the cap formula V=πh23(3R−h)V=\dfrac{\pi h^2}{3}(3R-h) to find the volume of a cap of height 22 from a sphere of radius 55.

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 π\pi.

Example 2

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

Example 3

easy
A sphere has radius 11. Find its volume in terms of π\pi.

Example 4

easy
A sphere has radius 66. Find its volume in terms of π\pi.

Example 5

easy
A sphere has diameter 66. Find its volume in terms of π\pi.

Example 6

easy
Use π≈3.14\pi\approx 3.14. A sphere has radius 33. Find its volume.

Example 7

medium
A hemisphere has radius 66. Find its volume in terms of π\pi.

Example 8

medium
A spherical scoop of ice cream has radius 22 cm. About how many cm³ is it? (Use π≈3.14\pi\approx 3.14.)

Example 9

medium
A sphere of radius 33 is melted and recast into spheres of radius 11. How many small spheres are formed?

Example 10

medium
A spherical tank holds 500π3\tfrac{500\pi}{3} m³ of water. Find its diameter.

Example 11

hard
A solid hemisphere of radius 66 sits on a cylinder of radius 66 and height 44. Find the total volume in terms of π\pi.

Example 12

hard
A solid sphere is divided into 8 equal-volume spherical sub-balls. If the original radius is 44, find the radius of each sub-ball.

Example 13

hard
A hollow ball has outer radius 55 and inner radius 44. Find the volume of material in terms of π\pi.

Example 14

hard
Earth's volume is roughly 1.083×10121.083\times 10^{12} km³. Estimate Earth's radius to 2 significant figures (use π≈3.14\pi\approx 3.14).
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Background Knowledge

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

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