Circles and a Square

The side length of the square which is also the diameter of the small circle is twice the radius of the small circle and it is $2 \times 3 = 6$ . The diameter of the big circle which is also the diagonal of the square can be found by using the pythagorean theorem and it is $6\sqrt{2}$ . Now that we know the diameter of the small and the big circle, we need to find their circumferences. The circumference of a circle is given by $c=\pi d$ where $d$ is the diameter of the circle. By using the formula, the circumference of the small circle is $6\pi$ and the big circle is $6\pi \sqrt{2}$ . To find the ratio of the circumference of the bog circle to the circumference of the small circle, we need to divide

$ratio=\dfrac{circumference~of~the~big~circle}{circumference~of~the~small~circle}=\dfrac{6\pi \sqrt{2}}{6\pi}=\sqrt{2}$

It is of the form $\sqrt{n}$ as required in the problem. The required answer is $\boxed{2}$ .