3. The perimeter of a square inscribed in a circle is $p$. What is the area of the square that circumscribes the circle?
3. $\frac{p^{2}}{8}$
The area of the square that circumscribes the circle is equal to the square of the diameter of the circle. The side of the inner square has length equal to $p / 4$, so that the diameter of the circle (which is equal to the length of the diagonal of the inner square) is given by
$$
\sqrt{\left(\frac{p}{4}\right)^{2}+\left(\frac{p}{4}\right)^{2}}=\frac{\sqrt{2} p}{4} \text {. }
$$