Now let the corners be rounded; each "corner" is an arc of a circle of radius r. If r grows to 1/[2*tan(pi/N)], the figure becomes a complete circle, with an area/perimeter ratio of 1/[4*tan(pi/N)].
Find r to maximize the area/perimeter ratio.
(It might be easier to start with small values of N, and see if a pattern emerges.)
Source: Extension of previous puzzle.