Answer
$(36\pi - 72) \text{ square units}$
Work Step by Step
Using the standard form $x^2+y^2=r^2$ (where $r$ is the radius) as basis, then $r^2=36 \implies r=6$.
The area $A$ of a circle is give by the formula $A=\pi{r^2}$ where $r=\text{radius}$
Hence, the area of the circle is:
$$A=\pi(6^2)=36\pi \text{ square units}$$
The area of the shaded region is equal to the difference between the area of the circle and the area of the square.
Let the side of the square be $a$.
The diagonal divides the square into two congruent right triangles, with the base and height equal to $a$.
The diagonal of the square is the diameter of the circle.
Since the radius is $6$ units, then the diameter is $12$ units.
Hence, the diagonal of the square is $12$ units long.
The area of a square is equal to the square of its side.
Using Pythagorean Theorem givesx_{y}:
\begin{align*}
\text{(base)}^2 + \text{(height)}² &= \text{(hypotenuse)}^2\\
a^2 + a^2 &= 12^2\\
2a^2 &= 144\\
a^2 &= 72\\
\end{align*}
Thus, the area of the square is $72 \text{ square units}$.
This means that the area of the shaded region is:
\begin{align*}
\text{Area of the shaded region} &= \text{area of the circle} - \text{area of the square}\\
&= (36\pi - 72) \text{ square units}
\end{align*}
