Hi DeuceJ! I'm a little tired, so forgive me if I make a mistake, but I think I can offer some help.

Here's a big hint: a parabola is the set of all points in the plane equidistant from a fixed point (the focus) and a fixed line (the directrix). So, the locus of points equidistant from the center of the square and any one side will be a parabola. Considering all sides, you will have four intersecting parabolas which will, in the center, close off R (see graph).

Finding the area of this region will involve some integration. To find equations for the parabolas, start with , where is the vertex and is the distance between the vertex and the focus or directrix; for the horizontal parabolas just interchange and .

Now, since the region is symmetric as you seem to have observed, consider one corner. The two parabolas that lie in that quadrant can be written as a function of , as long as you only worry about one half of the horizontal one. Find their intersection, and integrate along the appropriate intervals. Then remember to multiply by 4 to get the total area.