1966 AHSME Problems/Problem 4

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Make half of the square's side x. Now the radius of the smaller circle is x, so it's area is pi*x^2

Now find the diameter of the bigger circle. Since half of the square's side is x, the full side is 2x. Using the Pythagorean theorem, you get the diagonal to be 2sqrt2*x. Half of that is the radius, or xsqrt2. Using the same equation as before, you get the area of the larger circle to be 2x^2*pi. Putting one over the other and dividing, you get two as the answer: or

(B)