Let O be the center of the regular polygon (the center of the circumcenter), and let R be the distance from the O to the vertices.

The angle between two lines beginning in O and ending in two adjacent vertices is thus

, and from the sinus law we get that the area of a triangle formed by such two lines is given by

But there are n such triangles in the regular polygon. Using our last result this gives us

Furthermore, if we drop an altitude to the base in one of those isosceles triangles, we come up with

, and after subtitution in the expression for the area we get

.

Q.E.D