To determine the number of sides in a regular polygon based on its interior angle, you can use the following formula:

$n = \frac{360}{180 - \text{interior angle}}$

A regular polygon is defined as a geometric figure with all sides and angles equal. The interior angle refers to the angle formed within the polygon at each vertex. To find the number of sides, denoted as $n$, when the interior angle is known, we employ a formula derived from the fundamental properties of polygons.

First, let’s recall that the sum of the interior angles of a polygon with $n$ sides is given by the formula:

$\text{Sum of interior angles} = (n-2) \times 180^\circ.$

For a regular polygon, since each interior angle is the same, the measure of each interior angle can be expressed as:

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

With the interior angle known, we can set up the following equation:

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

Next, we rearrange this equation to isolate $n$:

1. Multiply both sides by $n$:

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

2. Expand the right side:

$\text{interior angle} \times n = 180n - 360.$

3. Rearrange the equation:

$\text{interior angle} \times n - 180n = -360.$

4. Factor out $n$:

$n(\text{interior angle} - 180) = -360.$

5. Finally, solve for $n$:

$n = \frac{360}{180 - \text{interior angle}}.$

This formula enables you to calculate the number of sides in the polygon. For instance, if the interior angle is $120^\circ$, you can substitute this value into the formula:

$n = \frac{360}{180 - 120} = \frac{360}{60} = 6.$

Thus, the polygon has $6$ sides, which identifies it as a hexagon.