Let:
- $r=$ radius of the circle,
- $a=$ side length of the square,
- $z=$ area of the circle.

We are told that the circumference of circle $C$ equals the perimeter of square $S$ :

$$
\(2 \pi r=4 a\) .
$$


Solve for $a$ :

$$
\(a=\frac{2 \pi r}{4}=\frac{\pi r}{2}\)
$$


The area of the circle is:

$$
\(z=\pi r^2\)
$$


We want the area of the square:

$$
\(\text { Area of square }=a^2=\left(\frac{\pi r}{2}\right)^2=\frac{\pi^2 r^2}{4}\)
$$


Since $\(r^2=\frac{z}{\pi}\)$, substitute:

$$
\(a^2=\frac{\pi^2}{4} \times \frac{z}{\pi}=\frac{\pi z}{4} .\)
$$


Final answer:

$$
\(\frac{\pi z}{4}\),
$$

which corresponds to option C .