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?

You have a cube of ice floating in a glass of water. The question is what fraction of the ice will be above the water line? Assume that the ice is not bobbing.

Most of you have probably heard the answer to this before. But please provide a proof or solution, along with your assumptions.