The surface area of a cone （圆锥的表面积）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 （扇形的面积） 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.