I came across this very interesting puzzle earlier today and after a little work, I think I have the answer (but it was an open question with no given answer so i can't check). Find the formula for the shaded area in the picture below from the circles' radii of a, b, c.
2. "RE: area between 3 circles"
In response to message #1
There is a straightforward solution using Heron's formula for a triangle with sides (a+b), (b+c) and (a+c), from which sectors of each circle are subtracted. The central angles can be found from the Law of Cosines in that triangle.