To solve this, let's break down the problem step by step.

Step 1: Relationship between the hexagon and the circle

The given regular hexagon is circumscribed around a circle, which means that the circle is inscribed in the hexagon. For a regular hexagon:

• The radius $r$ of the circle is equal to the apothem of the hexagon, which is also the distance from the center of the hexagon to the midpoint of its sides.

The area $A$ of a regular hexagon is given by:

$A = \frac{3\sqrt{3}}{2} s^2$

where $s$ is the side length of the hexagon.

Step 2: Use the area to find the side length of the hexagon

The problem states that the area of the hexagon is $36\sqrt{3}$ cm². So, set the formula for the area equal to this value:

$\frac{3\sqrt{3}}{2} s^2 = 36\sqrt{3}$

Simplify and solve for $s^2$:

$\frac{3}{2} s^2 = 36$

Multiply both sides by 2 to eliminate the fraction:

$3s^2 = 72$

Now divide by 3:

$s^2 = 24$

Taking the square root of both sides:

$s = \sqrt{24} = 2\sqrt{6}$

Step 3: Calculate the radius of the circle

For a regular hexagon, the radius $r$ of the circumscribed circle is equal to the side length $s$ of the hexagon. Therefore:

$r = 2\sqrt{6}$

Step 4: Area of the inscribed square (regular quadrilateral)

Now, we need to find the area of a square inscribed in the same circle. The diagonal of the square is equal to the diameter of the circle, which is $2r$.

Thus, the diagonal $d$ of the square is:

$d = 2r = 2 \times 2\sqrt{6} = 4\sqrt{6}$

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

$d = a\sqrt{2}$

Substituting $d = 4\sqrt{6}$:

$4\sqrt{6} = a\sqrt{2}$

Solving for $a$:

$a = \frac{4\sqrt{6}}{\sqrt{2}} = 4\sqrt{3}$

Step 5: Calculate the area of the square

The area $A_{\text{square}}$ of the square is:

$A_{\text{square}} = a^2 = (4\sqrt{3})^2 = 16 \times 3 = 48 \text{ cm}^2$

Thus, the area of the square inscribed in the circle is 48 cm².

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Would you like more details on any part of the solution, or do you have further questions?

Here are some related questions:

1. How do you calculate the area of other polygons inscribed in a circle?
2. What is the relationship between the radius of a circle and different polygons inscribed within it?
3. Can the same approach be applied to non-regular polygons?
4. What is the area of a triangle inscribed in the same circle?
5. How does the side length of a square change if the circle's radius doubles?

Tip: When working with inscribed and circumscribed polygons, it's important to understand the relationship between side lengths, radii, and areas for regular shapes.