The side of a regular decagon is 2 cm
Find the circumradius of the decagon.

Options

A. `1+sqrt5 cm`
B. `-1+sqrt5 cm`
C. `2+sqrt3 cm`
D. none of these

Answer

To find the circumradius of a regular decagon with a side length of 2 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of sides and side length**:
- A regular decagon has 10 sides (n = 10).
- The length of each side (s) is given as 2 cm.

2. **Use the formula for the circumradius**:
- The formula for the circumradius (R) of a regular polygon is given by:
\[ R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} \]
- Here, \( s \) is the side length and \( n \) is the number of sides.

3. **Substitute the values into the formula**:
- Substitute \( s = 2 \) and \( n = 10 \):
\[ R = \frac{2}{2 \sin\left(\frac{\pi}{10}\right)} \]

4. **Simplify the expression**:
- The 2 in the numerator and denominator cancels out:
\[ R = \frac{1}{\sin\left(\frac{\pi}{10}\right)} \]

5. **Calculate \( \sin\left(\frac{\pi}{10}\right) \)**:
- We know that \( \sin\left(18^\circ\right) = \sin\left(\frac{\pi}{10}\right) \).
- The value of \( \sin(18^\circ) \) is given by:
\[ \sin(18^\circ) = \frac{\sqrt{5} - 1}{4} \]

6. **Substitute this value back into the equation for R**:
- Now substituting this value into our expression for R:
\[ R = \frac{1}{\frac{\sqrt{5} - 1}{4}} = \frac{4}{\sqrt{5} - 1} \]

7. **Rationalize the denominator**:
- To rationalize \( \frac{4}{\sqrt{5} - 1} \), multiply the numerator and denominator by \( \sqrt{5} + 1 \):
\[ R = \frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{4(\sqrt{5} + 1)}{5 - 1} = \frac{4(\sqrt{5} + 1)}{4} \]
- This simplifies to:
\[ R = \sqrt{5} + 1 \]

8. **Final Answer**:
- Therefore, the circumradius of the decagon is:
\[ R = 1 + \sqrt{5} \text{ cm} \]