3320.21 – A Square Inside a Circle Inside a Square

Given a circle with radius rr, inscribe a square in it. Also circumscribe a square around it. What is the ratio of the perimeter of the outer square to the perimeter of the inner square?

Solution

Consult the figure. If we arrange that the two squares are in just the right relationship, then it is clear that if half of the outer edge is xx then the inner edge is x2x \sqrt{2}. The ratio of the perimeters then is

8x4x2=22=2\dfrac{8 x}{4 x \sqrt{2}} = \dfrac{2}{\sqrt{2}} = \sqrt{2}