## 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.

Solution

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