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What is the elaborated Complete Solution?

We are given that a regular octagon with an area of 48 is inscribed in a circle. The problem asks for the area of a regular hexagon inscribed in the same circle.

Let's break this down step by step:

### Step 1: Area Formula for Regular Polygons
The area \( A \) of a regular polygon inscribed in a circle of radius \( r \), with \( n \) sides, is given by the formula:
\[
A = \frac{1}{2} n r^2 \sin\left(\frac{2\pi}{n}\right)
\]
where:
- \( n \) is the number of sides,
- \( r \) is the radius of the circumscribed circle,
- \( \sin\left(\frac{2\pi}{n}\right) \) is the sine of the central angle.

For an octagon, \( n = 8 \), and for a hexagon, \( n = 6 \).

### Step 2: Set Up the Equation for the Octagon
We know the area of the octagon is 48. Using the formula for the area of a regular octagon, we can express this as:
\[
A_{\text{octagon}} = \frac{1}{2} \times 8 \times r^2 \times \sin\left(\frac{2\pi}{8}\right)
\]
Simplifying:
\[
A_{\text{octagon}} = 4r^2 \sin\left(\frac{\pi}{4}\right)
\]
Since \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), the equation becomes:
\[
48 = 4r^2 \times \frac{\sqrt{2}}{2}
\]
\[
48 = 2\sqrt{2} r^2
\]
Solving for \( r^2 \):
\[
r^2 = \frac{48}{2\sqrt{2}} = \frac{24\sqrt{2}}{2} = 12\sqrt{2}
\]

### Step 3: Set Up the Equation for the Hexagon
Now, we use the same area formula for a regular hexagon inscribed in the same circle. The area of a regular hexagon is:
\[
A_{\text{hexagon}} = \frac{1}{2} \times 6 \times r^2 \times \sin\left(\frac{2\pi}{6}\right)
\]
Simplifying:
\[
A_{\text{hexagon}} = 3r^2 \sin\left(\frac{\pi}{3}\right)
\]
Since \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), the equation becomes:
\[
A_{\text{hexagon}} = 3r^2 \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} r^2
\]
Substitute \( r^2 = 12\sqrt{2} \) from the previous step:
\[
A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times 12\sqrt{2} = 18\sqrt{6}
\]

### Step 4: Conclusion
The area of the regular hexagon is \( 18\sqrt{6} \). Thus, the correct answer is:
\[
\boxed{(b) \ 18\sqrt{6}}
\]

Would you like further details or have any questions?

Here are some related questions you might want to explore:
1. How does the area formula change if the polygon is not regular?
2. What is the relationship between the side lengths of the octagon and hexagon inscribed in the same circle?
3. How would the result change if the octagon's area were different?
4. What is the central angle of a regular polygon with \( n \) sides inscribed in a circle?
5. Can you derive the formula for the perimeter of a regular polygon inscribed in a circle?

**Tip:** Understanding the properties of regular polygons inscribed in a circle can help solve many geometry problems efficiently.

A regular octagon of area 48 is inscribed in a circle. If a regular hexagon is inscribed in the same circle, what would its area be?

Solve a geometry problem involving a regular octagon with an area of 48 inscribed in a circle. Learn how to calculate the area of a regular hexagon inscribed in the same circle using trigonometric formulas for regular polygons. Suitable for Grades 11-12.

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