If R is the radius of circumscribing circle of a regular polygon of n-sides,then R =

Options

A. `a/2sin(pi/n)`
B. `a/2cos(pi/n)`
C. `a/2cosec(pi/n)`
D. `a/2cosec(pi/n)`

Answer

To find the radius \( R \) of the circumscribing circle of a regular polygon with \( n \) sides, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Regular Polygon**: A regular polygon with \( n \) sides has all sides equal and all interior angles equal. The angle subtended at the center by each side is \( \frac{2\pi}{n} \).

2. **Identify the Central Angle**: The central angle \( A \) corresponding to each side of the polygon can be expressed as:
\[ A = \frac{2\pi}{n} \]

3. **Use the Sine Rule**: In a triangle formed by two radii and one side of the polygon, we can apply the sine rule. The side length \( a \) of the polygon can be expressed in terms of the radius \( R \) and the central angle \( A \):
\[ a = 2R \sin\left(\frac{A}{2}\right) = 2R \sin\left(\frac{\pi}{n}\right) \]

4. **Relate Side Length to Radius**: From the sine rule, we can express the radius \( R \) in terms of the side length \( a \):
\[ R = \frac{a}{2 \sin\left(\frac{\pi}{n}\right)} \]

5. **Final Expression for Radius**: Rearranging gives us the formula for the radius \( R \):
\[ R = \frac{a}{2 \sin\left(\frac{\pi}{n}\right)} \]

6. **Expressing in Terms of Cosecant**: Since \( \sin\left(\frac{\pi}{n}\right) = \frac{1}{\csc\left(\frac{\pi}{n}\right)} \), we can rewrite the formula as:
\[ R = \frac{a}{2} \csc\left(\frac{\pi}{n}\right) \]

### Final Result:
Thus, the radius \( R \) of the circumscribing circle of a regular polygon with \( n \) sides is given by:
\[ R = \frac{a}{2 \sin\left(\frac{\pi}{n}\right)} \]