Given a unit square in the plane, consider the set of all points such that the square subtends an angle of 90 degrees at the point. Such a point is shown in the figure below. Find the area of the region bounded by this set.

What if we ask for the area of the region bounded by the set of all points such that the square subtends an angle of 45 degrees? [Note that in this case, there are two different ways the square can subtend an angle of 45 degrees (see the figure below).]