mathschallenge.net

Problem

A grey square has a white circle inscribed and a grey square inscribed in the circle.

What fraction of the diagram is white?

Solution

Area of outside square = 2r 2r = 4r²
Area of inside square = ½ 4r² = 2r²
Area of circle = πr²

Area white = πr² 2r² = r²(π 2)

Hence, fraction of the diagram white = r²(π 2)/4r² = (π 2)/4 0.285

If another white circle was inscribed in the inside square, what fraction would be white now?
What would happen if white circles and grey squares were recursively inscribed? Would the white area tend towards a limit?