Suppose a regular polygon, which is an N-sided with equal side lengths S and similar angles at each corner. There is an inscribed circle to the polygon that has center C and barely touches the midpoint of every part. A line from C to the midpoint of a side is known as the apothem, and consider this apothem has length R.
If you cut the polygon along staright lines from each corner of the polygon to the center C, you can get a bunch of triangles, each with area (1/2)*(base)*(height). Note that every (base) has length S and the (height) is the length R of the apothem, and there are N such triangles. Therefore the total area of the polygon is N*(1/2)*S*R, which to say it another way is:
(1/2) (Circumference of the Polygon) * R