Anyway, back to my half-assed thoughts on the subject, the formula for finding the area is similar because they're both shapes, but by definition, a polygon has a set number of straight sides, while a circle is theoretically a perfect curve, which is why you actually find the area with a different formula than you do with a polygon.
And if you're going to treat it like a perfect curve with infinite "sides", what's the point in ever referring to it as a polygon?