Circle Areas

[Bert Bates]

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?

[Joel McNary]

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 ]

[Bert Bates]

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