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How to Find Volume and Surface Area of a Cone

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A cone has a circle or ellipse as its base, and looks like this:

Cone with circular base

This entry only covers cones with a circular base.

Surface Area of a Cone

Formula

Surface Area of a Cone

A = π r^{2} + π \cdot r \cdot s

where $r$ is the radius of the base, and $s$ is the length of the side surface.

Example 1

Find the surface area of a cone where the radius is $5 cm$ , and the sides have a length of $8 cm$ :

The description means that $r = 5$ and $s = 8$ . You can just insert the values directly into the formula for the surface area:

A = π \cdot 5^{2} + π \cdot 5 \cdot 8 \approx 204.1 {cm}^{2}

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Want to watch animated videos and solve interactive exercises about finding the volume of a cone?

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Volume of a Cone

When finding the volume of a cone, you’re basically determining how much water you can fit into the cone if you pour it in with the top pointing downwards.

Formula

Volume of a Cone

V = \frac{π r^{2} \cdot h}{3}

Volume of a cone

Example 2

Find the volume of a cone with a radius of $20 cm$ and a height of $30 cm$ . Write the answer in liters.

Here $r = 20$ and $h = 30$ . You insert the values into the formula for the volume:

V = \frac{π \cdot 2 0^{2} \cdot 30}{3} \approx 12 560 {cm}^{3}

For the units, you know from here that $1$ dm³ = $1$ L, so you have to convert from cm³ to dm³:

\begin{array}{l} 12 560 {cm}^{3} & = (12 560 \div 1000) {dm}^{3} \\ = 12.56 {dm}^{3} \end{array}

\begin{array}{l} 12 560 {cm}^{3} = (12 560 : 1000) {dm}^{3} = 12.56 {dm}^{3} \end{array}

You have now determined that the cone holds

12.56 {dm}^{3} = 12.56 L

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