I did some calculations.... that seems to be the right answer! Not a proof or anything but here it is:
Take a washer w/ outer radius = 2 and inner radius = 1. The area = 2^2*pi - 1^2*pi = 3pi. Make a triangle w/ the chord and the center and cut it in half to make 2 right trianlges. Use good ol' a^2 + b^2 = c^2 and you get the chord = 2*sqrt(3).
Now take a random outer radius... lets say 4. Doing pretty much the same thing, you get an inner radius of sqrt(13). Then, 4^2*pi -sqrt(13)^2 * pi = 3pi.
Also works for outer radius of 5, but haven't tried different chord lengths.