Sure!

A regular polygon has "n" sides.

Therefore it may be subdivided into "n" isosceles triangles, all meeting at the centre.

The angle at the central apex of each triangle is 360/n degrees.

Subtract that from 180 degrees and divide by 2 to get the other angles "A" of each isosceles triangle.

Take half the length of a side of the polygon.

Use TanA multiplied by half a side length squared to get a triangle area.

"n" of these is the polygon area.