For each relationship below, find the radius of the circle that would cover exactly half of a one unit square.

A. The circle is entirely inside the square.

B. The center of the circle is at the corner of the square.

C. The center of the circle is at the midpoint of the side of the square.

D. The center of the circle is on the side of the square and the one corner of the square is on the circle.

Are there any interesting relationships between any of the radii above?

A and B are straightforward: 1/(sqrt(2pi)) and sqrt(2/pi) giving a ratio of 1:2. Other than that, not so easy.

The radii observed are:

0.39894..., 0.79788...,0.58282..., 0.59622...

Posted by broll on 2018-12-18 23:48:59