Imagine a circle: Now imagine a square inside the circle, such that all four of the square's corners touch the edge of the circle (inscribed). Now imagine a second circle inside the square,such that the inner circle touches the square at all four of the square's midpoints (inscribed). What's the ratio of areas of the two circles?

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I get 2:1 If the square has a length of 2x, then the inner circle has a radius of x and the outer circle a radius of x*sqrt(2). Apply A = PI * r^2 to each circle, and you get areas of PI * x^2 and * PI * x^2, respectively. Reduce (and express in terms of larger:smaller), and you get 2:1. [ November 16, 2003: Message edited by: Joel McNary ]

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Bert Bates
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I guess I'm gonna have to make these a bit harder...