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.

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.


Now for a cone problem:


Here’s the diagram 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


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:


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