2. A square is inscribed in a circle, as shown in the figure. If the area of the circle is $16 \pi \mathrm{cm}^{2}$ and the area of the square is $S \mathrm{~cm}^{2}$, what is the value of $\mathrm{S}$ ?
Solution: The area of a circle with radius $r$ is $\pi r^{2}$. Since this circle has area $16 \pi$, we have $r^{2}=16$, and $r=4$. So the circle has radius 4 and diameter 8 .

Let $t$ be the side length of the square. By the Pythagorean Theorem, $t^{2}+t^{2}=8^{2}$ so $t^{2}=32$, which is the area of the square.