Question

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

Answer 1

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:

\[
\boxed{6}
\]