A circle and a square have the same area. Find the length of the diagonal of the square divided by the radius of the circle. (Hint: draw a diagram and label unknowns.)
Since they have the same area we know.
$\displaystyle A=\pi r^2 \mbox{ and } A=s^2 \iff \pi r^2=s^2 \iff r^2=\frac{s^2}{\pi}$
$\displaystyle r=\frac{s\sqrt{2}}{\sqrt{\pi}}$
Now using the pythagorean theorem we know the length of the diagonal is
$\displaystyle d^2=s^2+s^2 \iff d^2 =2s^2 \iff d=s\sqrt{2}$
Now we are asked to find the ratio of diagonal to the radius
$\displaystyle \frac{s\sqrt{2}}{\frac{s\sqrt{2}}{\sqrt{\pi}}}=\sq rt{\pi}$