Let's analyze the problem by carefully examining the geometry of the regular hexagon and its relationship with the unit circle.

Step 1: Understanding the Placement

• A regular hexagon has six sides of equal length.
• One vertex of the hexagon is at the center of the unit circle.
• Two vertices lie on the circumference of the unit circle.
• One vertex lies outside of the unit circle.

Step 2: Geometry of the Hexagon and the Unit Circle

We can label the vertices of the hexagon as $A, B, C, D, E, F$ in a counterclockwise direction, with $A$ at the center of the unit circle. $B$ and $F$ are on the circumference of the unit circle. The length of side $AB = 1$ since it is a unit circle.

Step 3: Length of the Hexagon's Sides

Since $AB = 1$, the side length of the hexagon is 1 unit. Because it's a regular hexagon, all sides are equal to 1 unit.

Step 4: Hexagon Area Calculation

The area $A$ of a regular hexagon can be calculated using the formula: $A = \frac{3\sqrt{3}}{2} s^2$ where $s$ is the side length of the hexagon.

For $s = 1$: $A = \frac{3\sqrt{3}}{2} \times 1^2 = \frac{3\sqrt{3}}{2}$

However, this area corresponds to the hexagon where all vertices lie on the unit circle. But here, only two vertices are on the circle, so we need to reconsider the area of the described hexagon.

Step 5: Consideration of the Specific Configuration

Given that one vertex is at the center and two vertices lie on the circumference, we need to consider this specific arrangement.

The hexagon is divided into six equilateral triangles, with one of these triangles having its vertex at the center. The area of this particular triangle is $\frac{\sqrt{3}}{4} \times 1^2 = \frac{\sqrt{3}}{4}$.

The other five triangles each have a side length of 1, and their total area is calculated as: $5 \times \frac{\sqrt{3}}{4} = \frac{5\sqrt{3}}{4}$

Step 6: Total Area Calculation

Thus, the total area of the hexagon is the sum of these areas: $\text{Total Area} = \frac{\sqrt{3}}{4} + \frac{5\sqrt{3}}{4} = \frac{6\sqrt{3}}{4} = \frac{3\sqrt{3}}{2}$

Conclusion

The area of the hexagon is $\boxed{\frac{3\sqrt{3}}{2}}$.

Would you like more details on any of these steps, or do you have any questions?

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Here are five related questions for further exploration:

1. How does the area change if the hexagon is inscribed in the unit circle?
2. What is the area of a regular hexagon inscribed in a circle of radius 2?
3. Can you derive the formula for the area of a regular polygon with $n$ sides inscribed in a unit circle?
4. What would the side length be for a hexagon where all vertices lie on a circle of radius $r$?
5. How does the area of the hexagon compare to that of the equilateral triangles formed by its vertices?

Tip: When dealing with regular polygons and circles, symmetry can simplify calculations significantly.