If r is the radius of inscribed circle of a regular polygon of n-sides ,then r is equal to

Options

A. `a/2cot(pi/(2n))`
B. `a/2cot(pi/(n))`
C. `a/2tan(pi/(n))`
D. `a/2cos(pi/(n))`

Answer

To find the radius \( r \) of the inscribed circle of a regular polygon with \( n \) sides, we can derive the formula step by step as follows:

### Step 1: Understand the Regular Polygon
A regular polygon with \( n \) sides has all sides and angles equal. The inscribed circle (incircle) touches each side of the polygon at its midpoint.

### Step 2: Draw the Polygon and the Incircle
Draw a regular polygon with \( n \) sides. Inscribe a circle within this polygon. Label the center of the circle as \( O \) and the points where the circle touches the sides as \( A, B, C, \ldots \).

### Step 3: Identify Triangle and Angles
Consider one of the triangles formed by the center \( O \), a vertex \( A \), and the midpoint of the side opposite to \( A \), which we can denote as \( D \). The angle \( \angle AOD \) is the central angle corresponding to the side \( AB \).

### Step 4: Calculate the Central Angle
The total angle around point \( O \) is \( 360^\circ \) or \( 2\pi \) radians. Since there are \( n \) sides, the central angle \( \angle AOB \) for each side is:
\[ \text{Central angle} = \frac{360^\circ}{n} = \frac{2\pi}{n} \]
Thus, the angle \( \angle AOD \) (which is half of the central angle) is:
\[ \angle AOD = \frac{1}{2} \cdot \frac{2\pi}{n} = \frac{\pi}{n} \]

### Step 5: Use Trigonometric Ratios
In triangle \( AOD \), we can use the cotangent function. The radius \( r \) (which is \( OD \)) is the adjacent side to angle \( AOD \), and the length of half the side \( AB \) (which we denote as \( \frac{a}{2} \), where \( a \) is the length of one side of the polygon) is the opposite side.

Using the cotangent function:
\[ \cot\left(\frac{\pi}{n}\right) = \frac{\text{adjacent}}{\text{opposite}} = \frac{r}{\frac{a}{2}} \]

### Step 6: Solve for \( r \)
Rearranging the equation gives:
\[ r = \frac{a}{2} \cot\left(\frac{\pi}{n}\right) \]

### Conclusion
Thus, the radius \( r \) of the inscribed circle of a regular polygon with \( n \) sides is:
\[ r = \frac{a}{2} \cot\left(\frac{\pi}{n}\right) \]