Given a rectangle $A$ of sides $L$ and $W$, such that $L$ is less than $W$, what is the average area of all circles entirely contained in $A$? To be more explicit, what is the average area of all circles of all radii with centers located inside of $A$ such that each circle intersects any given side of A at most once?
I used calculus to find the solution to this problem and got that this average area is equal to
$(4W-3L)(\pi L^2)(72W)^{-1} .$
However, I was wondering if there exists a simpler, geometric interpretation to this problem.
EDIT: Thanks to joriki, the 36 was corrected to be 72.