If each side of a regular dodecagon is 1 cm, find the area of the dodecagon.

Options

A. 12 sq. cm
B. `6+3sqrt3 sq.cm`
C. `12sqrt3 sq.cm`
D. none of these

Answer

To find the area of a regular dodecagon (a polygon with 12 equal sides), we can use the formula for the area of a regular polygon:

\[ \text{Area} = \frac{n \cdot s^2}{4 \cdot \tan(\frac{\pi}{n})} \]

Where:
- \( n \) is the number of sides (for a dodecagon, \( n = 12 \))
- \( s \) is the length of each side

Given that each side of the dodecagon is 1 cm, we can substitute \( n = 12 \) and \( s = 1 \) into the formula.

### Step 1: Substitute the values into the formula
\[ \text{Area} = \frac{12 \cdot (1)^2}{4 \cdot \tan(\frac{\pi}{12})} \]

### Step 2: Simplify the equation
\[ \text{Area} = \frac{12}{4 \cdot \tan(\frac{\pi}{12})} \]
\[ \text{Area} = \frac{3}{\tan(\frac{\pi}{12})} \]

### Step 3: Calculate \( \tan(\frac{\pi}{12}) \)
Using the half-angle formula for tangent:
\[ \tan(\frac{\pi}{12}) = \tan(15^\circ) = 2 - \sqrt{3} \]

### Step 4: Substitute \( \tan(\frac{\pi}{12}) \) back into the area formula
\[ \text{Area} = \frac{3}{2 - \sqrt{3}} \]

### Step 5: Rationalize the denominator
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \text{Area} = \frac{3(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]
\[ = \frac{3(2 + \sqrt{3})}{4 - 3} = 3(2 + \sqrt{3}) = 6 + 3\sqrt{3} \]

### Final Answer
Thus, the area of the dodecagon is:
\[ \text{Area} = 6 + 3\sqrt{3} \text{ cm}^2 \]