2014 UNCO Math Contest II Problems/Problem 5

Problem

(a) The White Rabbit has a square garden with sides of length one meter. He builds a square cucumber frame in the center by connecting each corner of the garden to the midpoint of a far side of the garden, going clockwise, as shown in the diagram. What is the area of the region that is enclosed in the inner square frame?

(b) Suppose that the White Rabbit builds his square cucumber frame by connecting each corner of the garden to a point a distance from the next corner, going clockwise, as shown in the diagram. Now what is the area of the region that is enclosed in the inner square frame?