It is well-known that the set of points P such that a line segment subtends an angle at P of measure α is a circular arc having that segment as a chord and the central angle the segment subtends is 2α (see diagram below).
When the angle is 90 degrees, the region is the shaded figure below. Consisting of four semi-circles with radii of 1/2, the area of the region is π/2.
When the angle is 45 degrees, the region is the more complicated. If a side of the square subtends the angle, then the points of interest are in the shaded area shown below.
If the diagonal of the square subtends the angle, then the shaded area below contains points of interest (the dashed arcs are points where the side of the square blocks the view of the diagonal).
Combining the regions above gives the figure below whose (shaded) area we wish to compute.
We divide the shaded area into three types of regions: red regions whose area totals that of a circle of radius sqrt(2)/2 [i.e. π/2], blue regions whose area totals that of a circle of radius 1 [i.e. π], and yellow regions whose area is that of three unit squares [i.e. 3]. This gives a total area of 3 + 3π/2 or approximately 7.71238...
