Alternating Squares


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

What fraction of the diagram is white?


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

therefore Area white = πr2 minus 2r2 = r2(π minus 2)

Hence, fraction of the diagram white = r2(π minus 2)/4r2 = (π minus 2)/4 approximately 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?

Problem ID: 119 (May 2003)     Difficulty: 2 Star

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