To find the area of the shaded region, we need to calculate the area of the circle and the area of the square inscribed in the circle, then subtract the area of the square from the area of the circle.
1. Determining the diameter and radius of the circle:
The given diameter of the circle is [tex]\(12 \sqrt{2}\)[/tex] millimeters.
The radius [tex]\(r\)[/tex] of the circle can be calculated by dividing the diameter by 2:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2} \, \text{millimeters}
\][/tex]
2. Calculating the area of the circle:
The formula for the area [tex]\(A\)[/tex] of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
Using the radius we have:
[tex]\[
A = \pi (6 \sqrt{2})^2 = \pi (36 \times 2) = 72 \pi \, \text{square millimeters}
\][/tex]
So, the area of the circle is [tex]\(72\pi \, \text{square millimeters}\)[/tex].
3. Determining the side length of the inscribed square:
In the square inscribed in the circle, the diagonal of the square is equal to the diameter of the circle. The side length [tex]\(s\)[/tex] of the square can be calculated using the property that in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the hypotenuse (diagonal of the square) is [tex]\(\sqrt{2}\)[/tex] times the side length:
[tex]\[
\text{Side length} = \frac{\text{Diagonal}}{\sqrt{2}} = \frac{12 \sqrt{2}}{\sqrt{2}} = 12 \, \text{millimeters}
\][/tex]
4. Calculating the area of the square:
The area [tex]\(A\)[/tex] of the square is given by the side length squared:
[tex]\[
A = s^2 = 12^2 = 144 \, \text{square millimeters}
\][/tex]
5. Finding the area of the shaded region:
The area of the shaded region is the area of the circle minus the area of the square:
[tex]\[
\text{Area of shaded region} = 72\pi - 144 \, \text{square millimeters}
\][/tex]
Therefore, the area of the shaded region is [tex]\(72\pi - 144 \, \text{square millimeters}\)[/tex], which matches the option [tex]\((72\pi - 144) \, \text{mm}^2\)[/tex].
