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
