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

In real life, you could use it to calculate the amount of icing in a cake decorating bag. Or the amount of meat inside a gyro.

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

## Volume of a Cone Formula

\[V=\frac{1}{3}\pi r^2 h\]

## How to Find the Volume of a Cone

- Find the radius of the cone.
- Find the height of the cone that is perpendicular to the base of the cone.
- Substitute the radius into the Volume of a Cone Formula.

Make sure to label your answer with cubic units!

## Examples

What is the volume?

*What is the radius of the cone?*

The diagram tells me that the radius of the cone is \(\green 5\, cm\).

*What is the height of the cone?*

The diagram tells me that the perpendicular height of the cone is \(\green 12\, cm\).

*What is the volume?*

When I substitute \(\green 5\) and \(\green 12\) into the volume formula, I get…

\[V=\frac{1}{3}\pi({\green 5})^2({\green 12})\]

I can simplify the exponent by multiplying \(\green 5\) by itself \(2\) times.

\[{\green 5}\times {\green 5}={\green 25}\]

So, that means…

\[V=\frac{1}{3}\pi({\green 25})({\green 12})\]

Next, I’ll multiply \(\green 25\) by \(\green 12\).

\[({\green 25})({\green 12})={\green 300}\]

So, the volume is…

\[V=\frac{1}{3}\pi({\green 300})\]

Then I’ll divide \(\green 300\) by \(3\) because dividing by \(3\) is the same as multiplying by \(\frac{1}{3}\).

\[{\green 300}\div 3 = {\green 100}\]

So, the simplified volume formula is…

\[V={\green 100}\pi\]

The volume is \(\green 100\pi \, cm^3\).

I could also find the decimal approximation of the volume by multiplying \(\green 100\) by \(3.14\).

\[\green V \approx 314\, cm^3\]

What is the volume?

*What is the radius of the cone?*

The diagram tells me that the diameter of the cone is \(\purple 3\, m\).

The radius of the cone is half of the diameter, so I can divide \(\purple 3\, m\) by \(2\) to find the radius.

\[{\purple 3}\div 2={\purple 1.5\}\]

The radius of the cone is \(\purple 1.5 \, m\).

*What is the height of the cone?*

The diagram tells me that the perpendicular height of the cone is \(\purple 3.6\, m\).

*What is the volume?*

When I substitute \(\purple 1.5\) and \(\purple 3.6\) into the volume formula, I get…

\[V=\frac{1}{3}\pi({\purple 1.5})^2({\purple 3.6})\]

I can simplify the exponent by multiplying \(\purple 1.5\) by itself \(2\) times.

\[{\purple 1.5}\times {\purple 1.5}={\purple 2.25}\]

So, that means…

\[V=\frac{1}{3}\pi({\purple 2.25})({\purple 3.6})\]

Next, I’ll multiply \(\purple 2.25\) by \(\purple 3.6\).

\[({\purple 2.25})({\purple 3.6})={\\purple 8.1}\]

So, the volume is…

\[V=\frac{1}{3}\pi({\purple 8.1})\]

Then I’ll write \(\purple 8.1\) as a fraction and multiply it by \(\frac{1}{3}\).

\[{\purple \frac{81}{10}}\times \frac{1}{3} = {\purple \frac{81}{30}}\]

\(\purple 81\) and \(\purple 30\) are both divisible by \(3\), so I can reduce the fraction.

\[{\purple \frac{81}{30}}={\purple {27}{10}\]

So, the simplified volume formula is…

\[V={\purple \frac{27}{10} }\pi\]

The volume is \(\purple \frac{27\pi }{10}\, m^3\).

I could also write a decimal approximation for the volume by multiplying \(\purple \frac{27}{10}\) by \(3.14\).

\[\purple V \approx 8.478 \, m^3\]