Originally Posted by

**Dormir** I'll start from the beginning. I was trying to find a general formula for the ratio of a circle to a regular polygon that fit's exactly inside it as part of another problem. So it is (pi*r^2) / .5*aP where aP is apothem*perimeter. The interior angle (theta) of any x sided polygon, in radians, is pi(x-2)/2x. Making a triangle out of the apothem, the radius, and half of a side (s), and half of theta as our angle, we see that a = r sin (.5*theta). s = 2 r cos (.5*theta). P = sx, so P = 2rxcos(.5*theta). Put those into the first equation and the ratio = pi / ( x sin (.5*theta )*cos(.5* theta)). Using the identities we get 2*pi / sin (theta). When the ratio is 1, when the polygon and circle have equal areas, we can multiply both sides by the bottom and get 2pi = x*sin( pi(x-2)/2x). I hope I that made sense and I didn't make a mistake.