The limit of the interior angle of a regular polygon of n sides as `n rarroo` is

Options

A. `pi`
B. `(pi)/(3)`
C. `(3pi)/(2)`
D. `(2pi)/(3)`

Answer

To find the limit of the interior angle of a regular polygon with \( n \) sides as \( n \) approaches infinity, we can follow these steps:

### Step 1: Understand the formula for the interior angle of a regular polygon
The formula for the interior angle \( A \) of a regular polygon with \( n \) sides is given by:
\[ A = \frac{(n-2) \times 180^\circ}{n} \]
This formula arises because the sum of the interior angles of an \( n \)-sided polygon is \( (n-2) \times 180^\circ \), and dividing by \( n \) gives the measure of each interior angle.

### Step 2: Convert degrees to radians
To work with radians, we can convert \( 180^\circ \) to radians:
\[ 180^\circ = \pi \text{ radians} \]
Thus, the formula in radians becomes:
\[ A = \frac{(n-2) \times \pi}{n} \]

### Step 3: Simplify the expression for the interior angle
We can simplify the expression for \( A \):
\[ A = \frac{n\pi - 2\pi}{n} = \pi - \frac{2\pi}{n} \]

### Step 4: Take the limit as \( n \) approaches infinity
Now, we want to find the limit of \( A \) as \( n \) approaches infinity:
\[ \lim_{n \to \infty} A = \lim_{n \to \infty} \left( \pi - \frac{2\pi}{n} \right) \]
As \( n \) approaches infinity, the term \( \frac{2\pi}{n} \) approaches \( 0 \). Therefore:
\[ \lim_{n \to \infty} A = \pi - 0 = \pi \]

### Conclusion
Thus, the limit of the interior angle of a regular polygon as \( n \) approaches infinity is:
\[ \pi \text{ radians} \]