Calculate the Surface Area of a Cone

By Mark Ryan

To calculate the surface area of a cone, you need to add the area of the cone’s base to its lateral area. (The lateral area is a fancy name for the area of the surface that connects the base to the peak; in other words, everything but the base.)

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

The following formula gives you the surface area of a cone.

geometry-cone-SA

Here’s a little more background on the lateral area:

The lateral area of a cone is one “triangle” that’s been rolled into a cone shape like a snow-cone cup (it’s only kind of a triangle because when flattened out, it’s actually a sector of a circle with a curved bottom side). Its area is

geometry-slant-height

Now for a cone problem:

geometry-cone-SAproblem

Here’s the proof diagram.

geometry-cone-proof

For the surface area, the only other thing you need is the slant height, l. The slant height is the hypotenuse of the 30-60-90 triangle, so it’s just twice the radius, which makes it

geometry-four-square-three

Now plug everything into the cone surface area formula:

geometry-SA-formula