The images show a circle with the equation $x^2 + y^2 = 484$, indicating a circle with a radius of $\sqrt{484} = 22$, centered at the origin. Inside the circle, there is a square whose vertices are inscribed on the circle.

The problem asks for the area of the shaded region, which is the area of the circle minus the area of the square.

Step 1: Find the area of the circle

The formula for the area of a circle is:

$A_{\text{circle}} = \pi r^2$

Given that the radius $r = 22$, we can calculate the area:

$A_{\text{circle}} = \pi (22)^2 = 484\pi \, \text{square units}$

Step 2: Find the area of the square

The square is inscribed in the circle, so its diagonal is equal to the diameter of the circle. The diameter of the circle is:

$d = 2 \times r = 2 \times 22 = 44$

The relationship between the side length $s$ of a square and its diagonal $d$ is given by:

$d = s\sqrt{2}$

Solving for $s$, we get:

$s = \frac{d}{\sqrt{2}} = \frac{44}{\sqrt{2}} = 44 \times \frac{\sqrt{2}}{2} = 22\sqrt{2}$

Thus, the area of the square is:

$A_{\text{square}} = s^2 = (22\sqrt{2})^2 = 22^2 \times 2 = 968 \, \text{square units}$

Step 3: Find the area of the shaded region

The shaded area is the area of the circle minus the area of the square:

$A_{\text{shaded}} = A_{\text{circle}} - A_{\text{square}} = 484\pi - 968$

Thus, the area of the shaded region is:

$\boxed{484\pi - 968} \, \text{square units}$

This matches the answer provided in the second image.

Let me know if you'd like more details or further clarifications!

Here are 5 related questions you may find interesting:

1. How would the area of the shaded region change if the square were replaced with a rectangle?
2. What happens if the circle's radius were doubled?
3. How could the formula for the area of a square inscribed in a circle be generalized to polygons with more sides?
4. How can we calculate the perimeter of the square inscribed in the circle?
5. How would the area of the shaded region change if the square was positioned outside the circle?

Tip: When dealing with geometric shapes inscribed in others, always look for relationships between radii, diameters, and side lengths!