How do you find the number of sides in a regular polygon given its interior angle?

To find the number of sides in a regular polygon given its interior angle, use the formula: \( n = \frac{360}{180 - \text{interior angle}} \).

A regular polygon is a shape with all sides and angles equal. The interior angle is the angle inside the polygon at each vertex. To find the number of sides (n) in a regular polygon when you know the interior angle, you can use a specific formula derived from the properties of polygons.

First, recall that the sum of the interior angles of a polygon with \( n \) sides is given by \( (n-2) \times 180^\circ \). For a regular polygon, each interior angle is the same, so the measure of each interior angle is \( \frac{(n-2) \times 180^\circ}{n} \).

Given the interior angle, you can set up the equation:
\[ \text{interior angle} = \frac{(n-2) \times 180^\circ}{n} \]

Rearrange this equation to solve for \( n \):
\[ \text{interior angle} \times n = (n-2) \times 180^\circ \]
\[ \text{interior angle} \times n = 180n - 360 \]
\[ \text{interior angle} \times n - 180n = -360 \]
\[ n(\text{interior angle} - 180) = -360 \]
\[ n = \frac{360}{180 - \text{interior angle}} \]

This formula allows you to calculate the number of sides in the polygon. For example, if the interior angle is 120 degrees, substitute it into the formula:
\[ n = \frac{360}{180 - 120} = \frac{360}{60} = 6 \]
So, the polygon has 6 sides, making it a hexagon.