A letter C is in the form of a sector from a unit circle from which a smaller concentric circle has been removed. The goal in each case is the have the center of gravity at the inner edge of the letter.

Case 1: Find the angle of the sector if the inner radius is .5

Case 2: Find the inner radius if the angle of the sectors is 270 degrees.

The center of gravity or centroid of a sector can be found using a formula given here:
http://mathworld.wolfram.com/CircularSector.html