You just need to do the simulation once. It's an elementary result that if the solution for a circle of radius 1 is k, the solution for a circle of radius 2 wil

You just need to do the simulation once. It's an elementary result that if the solution for a circle of radius 1 is k, the solution for a circle of radius 2 will be 2k. That was already done in the 19th century without computers. But like you say, that gives only a "close enough" formula. Close enough for farmers, or even rocket engineers who might need much more precision, but not good enough for mathematicians.