A regular hexagon is placed on top of a unit circle such that one vertex coincides with
the center of the circle, exactly two vertices lie on the circumference of the circle, and exactly
one vertex lies outside of the circle. Determine the area of the hexagon.

We are given a regular hexagon placed on top of a unit circle such that one vertex coincides with the center of the circle, two vertices lie on the circumference, and one vertex lies outside of the circle. The goal is to determine the area of this hexagon.

### Step 1: Analyze the Configuration
Let's denote:
- \( O \) as the center of the unit circle and one vertex of the hexagon.
- The hexagon is regular, so all six sides are equal, and the interior angles are \( 120^\circ \).
- Two of the hexagon's vertices lie on the circumference of the unit circle, meaning the distance from \( O \) (the center) to these vertices is \( 1 \) unit, since it is the radius of the unit circle.

### Step 2: Geometry of a Regular Hexagon
In a regular hexagon, the distance from the center to any vertex (circumradius \( R \)) is related to the side length \( s \) by the formula:
\[
R = s
\]
since for a regular hexagon, the radius is equal to the side length of the hexagon.

Given that two vertices of the hexagon lie on the circumference of the unit circle, the side length \( s \) of the hexagon must be \( 1 \).

### Step 3: Determine the Area of the Hexagon
The formula for the area \( A \) of a regular hexagon with side length \( s \) is given by:
\[
A = \frac{3\sqrt{3}}{2} s^2
\]
Substituting \( s = 1 \):
\[
A = \frac{3\sqrt{3}}{2} \times 1^2 = \frac{3\sqrt{3}}{2}
\]

### Conclusion:
The area of the hexagon is \( \frac{3\sqrt{3}}{2} \) square units.

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Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do you derive the formula for the area of a regular hexagon?
2. What is the relationship between the circumradius and side length in a regular polygon?
3. How does the placement of the hexagon affect its properties when placed on a unit circle?
4. Can the area of any polygon be determined in a similar way if vertices are on a circle?
5. How does the side length of the hexagon change if the radius of the circle is different?

**Tip**: When dealing with regular polygons, symmetry often simplifies complex problems, especially those involving circles.

A regular hexagon is placed on top of a unit circle such that one vertex coincides with the center of the circle, exactly two vertices lie on the circumference of the circle, and exactly one vertex lies outside of the circle. Determine the area of the hexagon.

Explore how to determine the area of a regular hexagon placed on a unit circle, where one vertex coincides with the center, two vertices lie on the circle's circumference, and one vertex is outside. This solution uses geometric principles and formulas for regular polygons and unit circles, ideal for students in Grades 10-12.

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