>>10899611Yes, there is (if one takes as given the formulae for circular segments/circular sectors):
-fix your figure in the Cartesian plane, identify important points (intersection points, I found it useful to find other points along the main diagonal axis)
-apply the circular segment formula at appropriate points
-bookkeeping, the pythagorean theorem a time or two as I recall, organize.
What comes out is some frightful thing with like arctangents and/or arcsines, or similar; it can be simplified from like 16 terms down to 8 terms or such. You can then (of course) reconcile this result with the one given by instead using calculus, establishing that the two things which are supposed to be equal, are. Then you pick the simpler of the two formulae and call it good.
https://en.wikipedia.org/wiki/Circular_segment 
