If a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square.

Given the area of the circle is $9\pi$, we can first find the radius of the circle using the formula for the area of a circle:

$\text{Area of circle} = \pi r^2$ $9\pi = \pi r^2$

Dividing both sides by $\pi$:

$r^2 = 9$

Taking the square root of both sides:

$r = 3$

Since the radius of the circle is 3, the diameter of the circle is:

$\text{Diameter} = 2 \times r = 2 \times 3 = 6$

The diameter of the circle is equal to the side length of the square. Therefore, the side length of the square is 6.

Now, the area of the square is:

$\text{Area of square} = \text{side}^2 = 6^2 = 36$

Thus, the area of the square is $36$.

Would you like more details or have any questions?

Here are 5 related questions for you to explore:

1. What is the perimeter of the square in this problem?
2. How do you find the circumference of the inscribed circle?
3. What would happen to the area of the square if the area of the circle doubled?
4. If the square had a different shape inscribed (like an equilateral triangle), how would you find its area?
5. How does the relationship between a circle and an inscribed polygon change with different shapes?

Tip: Always double-check the relationships between inscribed shapes—side lengths and diameters play a key role in calculations.