Apothems are lines from the center of the polygon to a side such that it creates a right angle. For example:

Regular polygons can be cut up into a bunch of right triangles.

The pentagon pictured above can be cut into 12 right triangles. Let a = length of apothem and s = length of side.

Each right triangle has area (1/2)(a)(s/2)=(as/4). Since there's 10 triangles, you multiply that by 10 to get (5as)/2.

In general, the area would be (as/4)(2n) with n being the number of sides.

I think that's how it works, lemme go google to verify myself xD