If a regular polygon of `n` sides has circum radius `R` and inradius `r` then each side of polygon is:

Options

A. `(R+r) tan ((pi)/(2n))`
B. `2(R+r)tan((pi)/(2n))`
C. `(R+r)sin((pi)/(2n))`
D. `2(R+r)cot((pi)/(2n))`

Answer

To find the length of each side of a regular polygon with \( n \) sides, circumradius \( R \), and inradius \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- A regular polygon can be inscribed in a circle (circumcircle) with radius \( R \) and can also be circumscribed by a circle (incircle) with radius \( r \).
- The angle subtended by each side at the center of the polygon is \( \frac{2\pi}{n} \).

2. **Finding the Half-Angle**:
- The angle at the center for half of one side is \( \frac{\pi}{n} \).

3. **Using Right Triangle Relationships**:
- Consider the triangle formed by the center of the polygon \( O \) and two adjacent vertices \( A \) and \( B \). The length of the side \( L \) can be expressed using the sine function:
\[ \frac{L}{2} = R \sin\left(\frac{\pi}{n}\right) \]
- Therefore, we can express \( L \) as:
\[ L = 2R \sin\left(\frac{\pi}{n}\right) \]

4. **Using the Inradius**:
- In a similar triangle formed with the inradius \( r \):
\[ \frac{L}{2} = r \tan\left(\frac{\pi}{n}\right) \]
- Thus, we can also express \( L \) as:
\[ L = 2r \tan\left(\frac{\pi}{n}\right) \]

5. **Equating the Two Expressions for \( L \)**:
- From the two equations for \( L \):
\[ 2R \sin\left(\frac{\pi}{n}\right) = 2r \tan\left(\frac{\pi}{n}\right) \]

6. **Simplifying the Equation**:
- Dividing both sides by 2:
\[ R \sin\left(\frac{\pi}{n}\right) = r \tan\left(\frac{\pi}{n}\right) \]
- Recall that \( \tan\left(\frac{\pi}{n}\right) = \frac{\sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)} \):
\[ R \sin\left(\frac{\pi}{n}\right) = r \frac{\sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)} \]
- Assuming \( \sin\left(\frac{\pi}{n}\right) \neq 0 \) (which is valid for \( n > 2 \)), we can divide both sides by \( \sin\left(\frac{\pi}{n}\right) \):
\[ R = r \sec\left(\frac{\pi}{n}\right) \]

7. **Finding Each Side of the Polygon**:
- Now, substituting back into the expression for \( L \):
\[ L = 2r \tan\left(\frac{\pi}{n}\right) \]
- Therefore, the length of each side of the polygon can be expressed as:
\[ L = r + R + r \tan\left(\frac{\pi}{2n}\right) \]

### Final Expression:
The length of each side of the polygon is given by:
\[ L = r + R + r \tan\left(\frac{\pi}{2n}\right) \]