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.

Read the full concept explanation →

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

\pi

Find the volume of a sphere with radius 12 cm

Answer

V = 2304\pi

cm³.

First step

Step 1: Write the formula:

V = \frac{4}{3}\pi r^3

Full solution

2
Step 2: Substitute $r = 12$ : $V = \frac{4}{3}\pi (12)^3 = \frac{4}{3}\pi \times 1728$ .
3
Step 3: Simplify: $\frac{4}{3} \times 1728 = 4 \times 576 = 2304$ . So $V = 2304\pi$ cm³.

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}

cm³. Find its radius.

Given V = 500π/3 cm³, find the radius

Example 3

medium

A solid sphere has volume

972\pi

. Find its radius.

Given V = 972π, find the radius of the sphere

Example 4

medium

A spherical ball just fits inside a cube of side

10

. Find the volume of the empty space inside the cube, in terms of

\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

3

is dropped into a cylinder of radius

3

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

2:3

Example 8

challenge

A spherical cap of height

h

is cut from a sphere of radius

R

. Use the cap formula

V=\dfrac{\pi h^2}{3}(3R-h)

to find the volume of a cap of height

2

from a sphere of radius

5

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

Sphere with diameter 10 cm (radius 5 cm) — find the volume

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

1

. Find its volume in terms of

\pi

Sphere with r = 1 — find volume in terms of π

Example 4

easy

A sphere has radius

6

. Find its volume in terms of

\pi

Sphere with r = 6 — find volume in terms of π

Example 5

easy

A sphere has diameter

6

. Find its volume in terms of

\pi

Sphere with diameter 6, radius 3 — find volume

Example 6

easy

Use

\pi\approx 3.14

. A sphere has radius

3

. Find its volume.

Example 7

medium

A hemisphere has radius

6

. Find its volume in terms of

\pi

Example 8

medium

A spherical scoop of ice cream has radius

2

cm. About how many cm³ is it? (Use

\pi\approx 3.14

Example 9

medium

A sphere of radius

3

is melted and recast into spheres of radius

1

. How many small spheres are formed?

Example 10

medium

A spherical tank holds

\tfrac{500\pi}{3}

m³ of water. Find its diameter.

Example 11

hard

A solid hemisphere of radius

6

sits on a cylinder of radius

6

and height

4

. Find the total volume in terms of

\pi

Example 12

hard

A solid sphere is divided into 8 equal-volume spherical sub-balls. If the original radius is

4

, find the radius of each sub-ball.

Example 13

hard

A hollow ball has outer radius

5

and inner radius

4

. Find the volume of material in terms of

\pi

Example 14

hard

Earth's volume is roughly

1.083\times 10^{12}

km³. Estimate Earth's radius to 2 significant figures (use

\pi\approx 3.14

← Back to Volume of a Sphere Formula Explained Practice Mode

Related Concepts

Area of a Circle Volume Pi (π)Sphere Surface Area Scaling in Space

Background Knowledge

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

area of circlevolumepi