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 \(3.14\).
\[\purple V \approx 8.478 \, m^3\]
