The volume of a sphere is the amount of three dimensional space inside of a sphere. Volume is a three dimensional measurement so it is measured in cubic units like \(in^3\), \(cm^3\), and \(ft^3\).

In real life, you could use it to calculate the amount of ice cream in a perfectly round ball of ice cream. Or the amount of water you would need to fill up a perfectly round water balloon.

Volume is the space inside of a sphere. If you are looking for the amount of flat space on the outside surface of a sphere, that is called the surface area of a sphere and it has a different formula than volume does.

## Volume of a Sphere Formula

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

## How to Find the Volume of a Sphere

- Find the radius of the sphere.
- Substitute the radius into the Volume of a Sphere Formula.

Make sure to label your answer with cubic units!

## Examples

These examples show how to find the volume when you are given the radius or diameter of a sphere. The examples on this page show how to use the Volume of a Sphere Formula to find the radius or diameter of a sphere if you are given the volume.

What is the volume?

*What is the radius of the sphere?*

The diagram tells me that the radius of the sphere is \(\green 6\, cm\).

*What is the volume?*

When I substitute \(\green 6\) into the volume formula, I get…

\[V=\frac{4}{3}\pi ({\green 6})^3\]

I can simplify the exponent by multiplying \(\green 6\) by itself \(3\) times.

\[{\green 6}\times {\green 6}\times {\green 6}={\green 216}\]

So, that means…

\[V=\frac{4}{3}\pi ({\green 216})\]

Next, I’ll write \(\green 216\) as a fraction and multiply the fractions.

\[\frac{4}{3}\times\frac{\green 216}{1}={\frac{\green864}{3}}\]

\(\green 864\) is divisible by \(3\), so I can reduce the fraction.

\[{\frac{\green 864}{3}}={\green 288}\]

So, the simplified volume formula is…

\[V={\green 288}\pi\]

The volume is \(\green 288\pi \, cm^3\).

I could find the decimal approximation of the volume by multiplying \(\green 288\) by \(3.14\).

\[\green V\approx 904.32\, cm^3\]

What is the volume?

*What is the radius of the sphere?*

The diagram shows me that the radius is \(\blue \frac{3}{4}\, in\).

*What is the volume?*

When I substitute \(\blue \frac{3}{4}\) into the volume formula, I get…

\[V=\frac{4}{3} \pi \left( {\blue \frac {3}{4}} \right )^3\]

I can simplify the exponent by multiplying \(\blue \frac {3}{4}\) by itself \(3\) times.

\[{\blue \frac {3}{4}}\times {\blue \frac {3}{4}}\times {\blue \frac {3}{4}}={\blue \frac {27}{64}}\]

So, that means…

\[V=\frac{4}{3} \pi \left ({\blue \frac {27}{64}} \right )\]

Next, I’ll multiply the fractions.

\[\frac{4}{3}\times{\blue \frac {27}{64}}={\blue \frac {108}{192}}\]

\(\blue 108\) and \(\blue 192\) are both divisible by \(12\), so I can reduce the fraction.

\[{\blue\frac{108}{192}}={\blue \frac {9}{16}}\]

So, the simplified volume formula is…

\[V={\blue \frac{9}{16}}\pi \]

The volume is \(\blue \frac{9\pi}{16} \, in^3\).

I could find the decimal approximation of the volume by multiplying \(\blue \frac{9}{16}\) by \(3.14\).

\[\blue V\approx 1.76625\, in^3\]

What is the volume?

*What is the radius of the sphere?*

The diagram shows me that the diameter of the sphere is \(\red 2.6\, m\).

The radius of the sphere is half of the diameter so I can divide \(\red 2.6\, m\) by \(2\) to find the radius.

\[{\red 2.6} \div 2 = {\red 1.3}\]

The radius of the sphere is \(\red 1.3 \, m\).

*What is the volume?*

When I substitute \(\red 1.3\) into the volume formula, I get…

\[V=\frac{4}{3}\pi \left( {\red 1.3} \right )^3\]

I can simplify the exponent by multiplying \(\red 1.3\) by itself \(3\) times.

\[{\red 1.3}\times {\red 1.3}\times {\red 1.3}={\red 2.197}\]

So, that means…

\[V=\frac{4}{3}\pi \left ({\red 2.197} \right )\]

Next, I’ll convert the decimal to a fraction and multiply the fractions.

\[\frac{4}{3}\times{\red \frac {2197}{1000}}={\red \frac {8788}{3000}}\]

\(\red 8788\) and \(\red 3000\) are both divisible by \(4\), so I can reduce the fraction.

\[{\red\frac{8788}{3000}}={\red \frac {2197}{750}}\]

So, the simplified volume formula is…

\[V={\red \frac{2197}{750}}\pi\]

The volume is \(\red \frac{2197\pi}{750} \, m^3\).

Or I could find a decimal approximation of the volume by multiplying \(\red \frac{2197}{750}\) by \(3.14\).

\[\red V\approx 9.1981\, m^3\]

## Why It Works

## Worksheets & Online Practice

Find the Volume when Given the Whole Number Radius of a Sphere

Answer Key Included

Find the Volume when Given the Decimal Radius of a Sphere

Answer Key Included