Ken's POTW

More Rounded Corners
In the preceding problem, we investigated a square. Now consider a regular polygon with N sides of length 1. The area/perimeter ratio of this figure is 1/[4*tan(pi/N)].

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.

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