Calculate the Volume of a Cone - dummies

By Mark Ryan

When you want to compute the volume of a cone, you need only two things: its height and the radius of its base. Even if you are given its slant height instead of its vertical height, you can still find the volume; you just need to include an extra step.

geometry-cone
A cone with its height and slant height.

A cone is a solid figure with a rounded base and a rounded lateral surface that connects the base to a single point.

A right circular cone is a cone with a circular base, whose peak lies directly above the center of the base. Most cones in geometry books are right circular cones.

Here’s how to find the volume of a cone.

geometry-cone-volume

Now for a cone problem:

geometry-cone-problem

Here’s the diagram proof.

geometry-cone-proof

To compute the cone’s volume, you need its height and the radius of its base. The radius is, of course, half the diameter, so it’s

geometry-square-three

Then, because the height is perpendicular to the base, the triangle formed by the radius, the height, and the slant height is a 30-60-90 triangle. You can see that h is the long leg and r the short leg, so to get h, you multiply r by the square root of 3:

geometry-h

You’re now ready to use the cone volume formula:

geometry-cone-volume-formula