What is the area of a regular hexagon inscribed in a circle of radius r ?

Options

A. `2sqrt(3) r^(2)` sq. units
B. `(3sqrt(3))/(2) r^(2)` sq. units
C. `(2)/(3) r^(2)` sq. units
D. `(sqrt(3))/(2) r^(2)` sq. units

Answer

To find the area of a regular hexagon inscribed in a circle of radius \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Hexagon**: A regular hexagon can be divided into 6 equilateral triangles. Each triangle has its vertices at the center of the circle and two adjacent vertices of the hexagon.

2. **Finding the Side Length**: The radius of the circle is equal to the length of each side of the equilateral triangles. Therefore, if the radius of the circle is \( r \), then the side length \( s \) of the hexagon is also \( r \).

3. **Area of One Equilateral Triangle**: The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting \( s = r \) into the formula, we get:
\[ A = \frac{\sqrt{3}}{4} r^2 \]

4. **Total Area of the Hexagon**: Since the hexagon consists of 6 such triangles, the total area \( A_{hex} \) of the hexagon is:
\[ A_{hex} = 6 \times A = 6 \times \frac{\sqrt{3}}{4} r^2 \]
Simplifying this gives:
\[ A_{hex} = \frac{6\sqrt{3}}{4} r^2 = \frac{3\sqrt{3}}{2} r^2 \]

5. **Final Result**: Therefore, the area of the regular hexagon inscribed in a circle of radius \( r \) is:
\[ A_{hex} = \frac{3\sqrt{3}}{2} r^2 \]