Let's break down the problem step-by-step.
1. Identify the diameter of the circle and calculate the radius:
The diameter of the circle is given as \(12 \sqrt{2}\) millimeters.
Calculation of the radius:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2} \text{ millimeters}
\][/tex]
2. Calculate the area of the circle:
The formula for the area of a circle is \(A = \pi r^2\), where \(r\) is the radius.
Calculation of the area:
[tex]\[
\text{Area of the circle} = \pi (6 \sqrt{2})^2 = \pi \times 72 = 72\pi \text{ square millimeters}
\][/tex]
3. Determine the side length of the inscribed square:
In an inscribed square, the diagonal of the square is equal to the diameter of the circle. For a square, \(s \sqrt{2}\) (the diagonal) can be equated to the given diameter.
Calculation of the side length:
[tex]\[
\text{Side length} = \frac{\text{Diagonal}}{\sqrt{2}} = \frac{12 \sqrt{2}}{\sqrt{2}} = 12 \text{ millimeters}
\][/tex]
4. Calculate the area of the square:
The area of a square is given by \(A = s^2\), where \(s\) is the side length.
Calculation of the area:
[tex]\[
\text{Area of the square} = 12^2 = 144 \text{ square millimeters}
\][/tex]
5. Calculate the area of the shaded region:
The shaded region is the area of the circle minus the area of the inscribed square.
Calculation of the shaded area:
[tex]\[
\text{Area of the shaded region} = \text{Area of the circle} - \text{Area of the square} = 72\pi - 144 \text{ square millimeters}
\][/tex]
Therefore, the area of the shaded region is \((72\pi - 144) \text{ mm}^2\). The correct answer choice is:
[tex]\[
\boxed{(72\pi - 144) \text{ mm}^2}
\][/tex]
