The?surface area of a cone?（圓錐的表面積）is equal to the curved surface area plus the area of the base:?

where:

?r?denotes the radius of the base of the cone,?

?L?denotes the slant height of the cone.?

The curved surface area is also called the lateral area.

Derivation of the Formula

In the cone below,?h?is the height,?L?is the slant height,?c?is the circumference of the base, and?r?is the radius of the base. How does it look when we unroll it?

If we unroll it, the shape is as follows:

It is a?sector of a circle?with radius?L?and arc length?c. So the curved surface area of the cone is the area of the sector above. The area of a sector （扇形的面積）?given the arc length?c?and radius?L?is given by

Now applying this to the cone, we have?A (curved) =cL/2,?where?L?is the slant height and?c?is the circumference of the base.?

After some manipulations,?

A (curved)?=πLr,?as given in the definition.