The number of sides of a regular polygon is given by:

Question

Accepted Answer

The correct option is 4:
The most direct and fundamentally correct method for determining the number of sides (n) of any regular polygon relies on the measure of its exterior angle. A key theorem in geometry states that the sum of the exterior angles of any convex polygon, regardless of the number of sides, is always 360°. In a regular polygon, all exterior angles are equal in measure. Therefore, if we let E represent the measure of a single exterior angle, the total sum is n × E = 360°. To find the number of sides (n), you simply rearrange this fundamental equation to $n = \frac{360^\circ}{\text{Exterior Angle}}$. This formula works because dividing the total measure of the exterior angles by the measure of a single, equal exterior angle tells you exactly how many angles (and thus sides) the polygon has.

The incorrect option is 1:
The formula $\frac{\text{Internal Angle}}{180^\circ}$ does not yield the number of sides of a regular polygon. The interior angle of a regular $n$-sided polygon is given by $\frac{(n-2)180^\circ}{n}$. Using a simple ratio of the interior angle to $180^\circ$ is geometrically baseless for calculating the number of sides, as it is a complex function of $n$. For instance, a square's interior angle is $90^\circ$, and $\frac{90^\circ}{180^\circ}$ equals $0.5$, which is clearly incorrect since a square has 4 sides. This option incorrectly assumes a direct proportional relationship that does not exist in polygon geometry.

The incorrect option is 2:
The option $\frac{\text{External Angle}}{180^\circ}$ is incorrect because it does not result in the number of sides. The correct formula relates the exterior angle to $360^\circ$, not $180^\circ$. The sum of the exterior angles is $360^\circ$, representing a full rotation, while $180^\circ$ relates to the linear pair formed by the interior and exterior angles. For a regular octagon, the exterior angle is $45^\circ$. Applying this incorrect formula gives $\frac{45^\circ}{180^\circ} = 0.25$, but the polygon has 8 sides. Therefore, this calculation holds no geometrical meaning for finding $n$.

The incorrect option is 3:
The formula $\frac{\text{Internal Angle}}{360^\circ}$ is mathematically unsound for finding the number of sides, $n$. The interior angle is the angle *inside* the polygon, and its value, when divided by $360^\circ$, yields a fraction that is completely unrelated to the side count. For example, a pentagon has an interior angle of $108^\circ$. $\frac{108^\circ}{360^\circ}$ simplifies to $0.3$, not 5. This method is flawed because the sum of interior angles varies based on $n$, but the sum of *exterior* angles is fixed at $360^\circ$, which is the value needed for the divisor.