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.

Solutions were received from Bill Chapp, Radu Ionescu, and Nick Baxter.

A/P = (1-(4-pi)r^2) / ( 4-(8-2pi)r )

After differentiating and setting equal to zero, we find:

r = [A/P](r) = 1/[2+sqrt(pi)]

Both the radius and the ratio have the same value.
This scales with the size of the square. So if the side length is s,

r = [A/P](r) = s/[2+sqrt(pi)]

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