A square is inscribed in a circle of diameter [tex][tex]$12 \sqrt{2}$[/tex][/tex] millimeters. What is the area of the shaded region?

Recall that in a [tex][tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex][/tex] triangle, if the legs each measure [tex][tex]$x$[/tex][/tex] units, then the hypotenuse measures [tex][tex]$x \sqrt{2}$[/tex][/tex] units.

A. [tex][tex]$(72 \pi - 144) \text{ mm}^2$[/tex][/tex]
B. [tex][tex]$(72 \pi - 72) \text{ mm}^2$[/tex][/tex]
C. [tex][tex]$(288 \pi - 288) \text{ mm}^2$[/tex][/tex]
D. [tex][tex]$(288 \pi - 144) \text{ mm}^2$[/tex][/tex]



Answer :

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].