Ken's POTW

Rounded Corners
Consider a square of side 1. The area/perimeter ratio is 1/4. Now let the corners be rounded; each "corner" is a quarter circle of radius r. If r = 1/2, the square becomes a circle with an area/perimeter ratio of 1/4. Find r to maximize the area/perimeter ratio.

Source: rec.puzzles, citing Ingenious Mathematical Problems and Methods, L.A. Graham, Dover, 1959.

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