The area of a regular hexagon circumscribed around a circle is 36√3 cm². Find the area of a regular quadrilateral inscribed in this circle.

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.

This problem explores the relationship between a regular hexagon circumscribed around a circle and a square inscribed in the same circle. Given the hexagon's area of 36√3 cm², learn how to calculate the area of the inscribed square step-by-step using geometric formulas. Suitable for Grades 9-11.

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