To solve this problem, let’s break down the information given and relate it to the geometry of the circle and the square inscribed in it.

1. Circumference of Circle: We know that the circumference of the circle is $4\pi$.

   • The formula for the circumference of a circle is $C = 2\pi r$, where $r$ is the radius.
   • Setting $2\pi r = 4\pi$, we can solve for $r$: $r = \frac{4\pi}{2\pi} = 2$
   • So, the radius $r$ of the circle is 2.

2. Diameter of Circle: Since the radius is 2, the diameter $d$ (which is twice the radius) is: $d = 2 \times 2 = 4$

3. Relationship between Circle and Square: The square $ABCD$ is inscribed in the circle, meaning that the diameter of the circle is also the diagonal of the square.

4. Diagonal of the Square and Side Length:

   • For a square with side length $s$, the diagonal $d$ can be calculated by the Pythagorean theorem: $d = s\sqrt{2}$.
   • We know the diagonal $d = 4$, so we can set up the equation: $s\sqrt{2} = 4$ $s = \frac{4}{\sqrt{2}} = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}$
   • Therefore, the side length $s$ of the square is $2\sqrt{2}$.

5. Area of the Square: The area $A$ of a square is given by $A = s^2$.

   • Substituting $s = 2\sqrt{2}$: $A = (2\sqrt{2})^2 = 4 \times 2 = 8$

Final Answer:

The area of square $ABCD$ is $8$.

Would you like a more detailed breakdown of any part of this solution, or do you have any questions?

Here are some additional related questions to consider:

1. If the radius of the circle was 3 instead, what would the area of the inscribed square be?
2. How would the solution change if the square was circumscribed around the circle instead of inscribed?
3. What is the relationship between the area of a circle and the area of an inscribed square?
4. How do you calculate the area of an equilateral triangle inscribed in a circle of known radius?
5. Can you determine the area of a rectangle inscribed in a circle given only the circle's radius?

Tip: When working with inscribed shapes, using known geometric relationships (like the diagonal of a square inside a circle) helps simplify calculations.