smc

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 $2\sqrt{2}x$. Half of that is the radius, or $x\sqrt{2}$. 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 $\boxed{2}$.