A regular dodecagon is inscribed in a circle of radius 1 cm. find the area of the dodecagon.

Options

A. 3 sq. cm
B. `root(3)(2) sq. cm`
C. `root(2)(3) sq. cm`
D. none of these

Answer

To find the area of a regular dodecagon (12-sided polygon) inscribed in a circle of radius 1 cm, we can follow these steps:

### Step 1: Understand the Geometry
A regular dodecagon can be divided into 12 isosceles triangles, each having a vertex at the center of the circle and the base as one side of the dodecagon.

### Step 2: Calculate the Central Angle
The central angle for each triangle can be calculated as:
\[ \text{Central Angle} = \frac{360^\circ}{12} = 30^\circ \]

### Step 3: Area of One Triangle
The area \( A \) of one of these isosceles triangles can be calculated using the formula:
\[ A = \frac{1}{2} \times r^2 \times \sin(\theta) \]
where \( r \) is the radius of the circle (1 cm) and \( \theta \) is the central angle in radians. First, we convert \( 30^\circ \) to radians:
\[ 30^\circ = \frac{\pi}{6} \text{ radians} \]
Now, substituting the values:
\[ A = \frac{1}{2} \times (1)^2 \times \sin\left(\frac{\pi}{6}\right) \]
Since \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \):
\[ A = \frac{1}{2} \times 1 \times \frac{1}{2} = \frac{1}{4} \text{ cm}^2 \]

### Step 4: Total Area of the Dodecagon
Since there are 12 such triangles in the dodecagon, the total area \( A_{total} \) is:
\[ A_{total} = 12 \times A = 12 \times \frac{1}{4} = 3 \text{ cm}^2 \]

### Conclusion
Thus, the area of the regular dodecagon inscribed in a circle of radius 1 cm is:
\[ \boxed{3 \text{ cm}^2} \]