If an interior angle of a regular polygon measures 120°, how many sides does the polygon have?

1 answer

To find the number of sides of a regular polygon given the measure of its interior angle, we can use the formula for the interior angle of a regular polygon:

\[ \text{Interior angle} = \frac{(n-2) \times 180}{n} \]

where \( n \) is the number of sides of the polygon.

Given that the interior angle measures 120°, we can set up the equation:

\[ 120 = \frac{(n-2) \times 180}{n} \]

To eliminate the fraction, multiply both sides of the equation by \( n \):

\[ 120n = (n-2) \times 180 \]

Expanding the right side:

\[ 120n = 180n - 360 \]

Next, we can rearrange the equation to isolate \( n \):

\[ 120n - 180n = -360 \]

This simplifies to:

\[ -60n = -360 \]

Dividing both sides by -60 gives:

\[ n = 6 \]

Thus, the polygon has \( 6 \) sides.

To confirm, we can check the interior angle of a polygon with 6 sides (a hexagon):

The interior angle of a hexagon is calculated as follows:

\[ \text{Interior angle} = \frac{(6-2) \times 180}{6} = \frac{4 \times 180}{6} = \frac{720}{6} = 120° \]

Since this matches the given interior angle, we can conclude that the polygon is indeed a regular hexagon.

The final answer is:

\[ \boxed{6} \]