The interior angle of a regular polygon is :

Question

Accepted Answer

To determine the interior angle of a regular polygon, we need to understand the formula for calculating the interior angle based on the number of sides, $ n $, of the polygon. The sum of the interior angles of a polygon with $ n $ sides is given by the formula:

$$
(n-2) \times 180^\circ
$$

This formula arises because a polygon can be divided into $ n-2 $ triangles, and each triangle has an angle sum of $ 180^\circ $.

For a regular polygon, all interior angles are equal. Therefore, to find the measure of one interior angle, we divide the total sum of the interior angles by the number of sides, $ n $:

$$
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
$$

This matches Option 3, which is:

$$
\frac{(n-2) \times 180^\circ}{n}
$$

Now, let's explain why the other options are incorrect:

1. **Option 1: $(n-2) \times 180^\circ$**

This formula gives the sum of all interior angles of the polygon, not the measure of a single interior angle. Therefore, it is incorrect for finding the interior angle of a regular polygon.

2. **Option 2: $\frac{n}{2} \times 180^\circ$**

This formula does not correctly represent any known calculation related to the interior angles of a polygon. It neither gives the sum of the interior angles nor the measure of a single interior angle. Hence, it is incorrect.

3. **Option 4: $\frac{n(n-3)}{2}$**

This formula is unrelated to the calculation of interior angles. It resembles the formula for the number of diagonals in a polygon, which is $\frac{n(n-3)}{2}$, and has no relevance to the interior angles. Thus, it is incorrect for this context.

In conclusion, Option 3 is the correct choice because it accurately calculates the measure of an interior angle of a regular polygon by dividing the total sum of the interior angles by the number of sides.