Covering a polygon with n circular rings
This problem might be hard to solve exactly.  I would suggest you try gradient descent (or another mathematical optimization technique).



[ The stuff below here is obsolete, now that the question has been edited to change the problem statement along these lines. ]

Note that minimizing your error ratio is equivalent to minimizing

$$\text{Area}(S-R) + \text{Area}(R-S),$$

since $S$ is given (so $\text{Area}(S)$ is fixed).  This observation does not seem to help very much, though.