Given:
- A square is inscribed in a circle.
- The area of the circle is $18 \pi$.

We are asked to find the area of the square.

Step 1: Find the radius of the circle
The area of a circle is:

$$
\(\pi r^2=18 \pi\)
$$


Divide both sides by $\(\pi\)$ :

$$
\(\begin{gathered}
r^2=18 \\
r=\sqrt{18}=3 \sqrt{2}
\end{gathered}\)
$$


Step 2: Relationship between the circle and the square
- The square is inscribed in the circle $\rightarrow$ the circle circumscribes the square.
- The diameter of the circle equals the diagonal of the square.

Diameter of the circle:

$$
\(d=2 r=2 \times 3 \sqrt{2}=6 \sqrt{2}\)
$$


This is the diagonal of the square.

Step 3: Find the side length of the square
For a square with side length $s$,

$$
\(\text { diagonal }=s \sqrt{2}\)
$$


Set diagonal equal to circle diameter:

$$
\(s \sqrt{2}=6 \sqrt{2}\)
$$


Divide both sides by $\(\sqrt{2}\)$ :

$$
\(s=6\)
$$


Step 4: Find the area of the square

$$
\(\text { Area }=s^2=6^2=36\)
$$


Final answer:

\(36\)