Rhombus and Circular Pool

Mr. X has a rhombus shaped backyard. He wants to build the biggest possible circular pool in his backyard. How he could do it?

Extension 1
Construct the biggest possible circular pool if the shape of the backyard is an n-sided regular polygon (square, pentagon, hexagon, etc.).

Extension 2
Devise a plan to construct the biggest possible circular pool for any given shaped backyard.