To determine the area of the shaded region formed by a square inscribed in a circle, we need to follow these steps:
1. Determine the diameter of the circle:
Given in the problem, the diameter of the circle is [tex]\( 12\sqrt{2} \)[/tex] millimeters.
2. Find the side length of the square:
Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle. In a 45°-45°-90° triangle (which forms half of the square), if the legs each measure [tex]\( x \)[/tex] units, then the hypotenuse measures [tex]\( x\sqrt{2} \)[/tex]. Here, the hypotenuse [tex]\( x\sqrt{2} \)[/tex] is the diagonal of the square and is equal to the diameter of the circle:
[tex]\[
x\sqrt{2} = 12\sqrt{2} \implies x = 12
\][/tex]
Therefore, the side length of the square is [tex]\( 12 \)[/tex] millimeters.
3. Calculate the area of the square:
The area of the square is given by the side length squared:
[tex]\[
\text{Area of the square} = 12^2 = 144 \text{ mm}^2
\][/tex]
4. Calculate the radius of the circle:
The radius of the circle is half of its diameter:
[tex]\[
\text{Radius} = \frac{12\sqrt{2}}{2} = 6\sqrt{2} \text{ mm}
\][/tex]
5. Calculate the area of the circle:
The area of the circle is given by:
[tex]\[
\text{Area of the circle} = \pi \times (\text{Radius})^2 = \pi \times (6\sqrt{2})^2 = \pi \times 72 = 72\pi \text{ mm}^2
\][/tex]
6. Determine the area of the shaded region:
The shaded region is the area of the circle minus the area of the square:
[tex]\[
\text{Area of the shaded region} = 72\pi \text{ mm}^2 - 144 \text{ mm}^2
\][/tex]
Therefore, the area of the shaded region is [tex]\( 72\pi - 144 \text{ mm}^2 \)[/tex].
Thus, the answer is [tex]\((72\pi - 144) \text{ mm}^2\)[/tex].
