Problem of overlapping circles

[standardflop]

A friend asked me the following question: Two circles with radii $R$ and $r$ are placed so that the one with radius $r$ has its center on the circumference of the circle with radius $R$. How big should $r$ be, so that the area of the overlap is exactly $\pi R^2/2$.

The simple solution would be to insert a coordinate system and integrate over each of the circle equations (the intersection begin $r_0$). But the resulting equation of the form

$$\frac{\pi R^2}{2}=\int_0^{r_0}circle2+\int_{r_0}^r circle1$$
turns out to be quite difficult to solve for $r$ (due to trignometric terms).

Another approach would be the geometrical, and one would find a general result similar to equation (14) of http://mathworld.wolfram.com/Circle-CircleIntersection.html . My problem is just that this also gives an equation (again with trigonometric terms) which is hard to solve.

Does anyone have an idea which could give an analytic result (or, is it even possible)?

All the best.
-stdflp.

 

[EnumaElish]

How close does the A(R',r') formula get you to the analytic solution you need? What additional steps do you need to go through?

 

[standardflop]

The A-formula gives me an equation containing $cos^{-1}$ terms, which neither me or maple can solve. The integral-method is likewise not solveable for me.

A friend of mine told me that the problem has no simple analytic solution (squre root, fraction, etc.) to describe the relation of $r$ and $R$. But i am not sure of this.