If a square is inscribed in a circle that, in turn, is inscribed in a

Look at the diagram:
You can see that a diameter of the circle equals to a diagonal of the smaller square and a side of the bigger square: \(diameter=diagonal_{small}=side_{big}\);

Now, if the side of the bigger square is \(a\) then its perimeter will be \(P=4a\);
Next, a side of a smaller square will be \(\frac{a}{\sqrt{2}}\) and its perimeter \(p=\frac{4a}{\sqrt{2}}\);

\(\frac{P}{p}=\sqrt{2}\)

Answer: C.

Or as soon as you realize that the sides of bigger and smaller squares are \(a\) and \(\frac{a}{\sqrt{2}}\) respectively, so their ration is \(\sqrt{2}\), then as P=4*side then the ratio of the sides will be the same as the ratio of the perimeters, so the same \(\sqrt{2}\).