Question

In a regular polygon each interior angle is greater by 120 degrees than each exterior angle. Calculate the number of sides of the polygon

Answer 1

Let \( n \) denote the number of sides of the regular polygon. The formula for the measure of each exterior angle \( E \) of a regular polygon is given by:

\[
E = \frac{360}{n}
\]

The measure of each interior angle \( I \) can be calculated as:

\[
I = 180 - E = 180 - \frac{360}{n}
\]

According to the problem statement, each interior angle is greater by 120 degrees than each exterior angle, which gives us the equation:

\[
I = E + 120
\]

Substituting the expressions for \( I \) and \( E \):

\[
180 - \frac{360}{n} = \frac{360}{n} + 120
\]

To solve for \( n \), we will first simplify the equation:

1. Combine like terms:

\[
180 - 120 = \frac{360}{n} + \frac{360}{n}
\]

which simplifies to:

\[
60 = \frac{720}{n}
\]

2. Multiply both sides by \( n \):

\[
60n = 720
\]

3. Divide both sides by 60:

\[
n = \frac{720}{60} = 12
\]

Thus, the polygon has \( n = 12 \) sides.

To verify, we can check the measures of the interior and exterior angles.

- Each exterior angle:
\[
E = \frac{360}{12} = 30 \text{ degrees}
\]

- Each interior angle:
\[
I = 180 - E = 180 - 30 = 150 \text{ degrees}
\]

Now, we can verify the condition that the interior angle is greater by 120 degrees than the exterior angle:

\[
150 = 30 + 120
\]

This confirms the condition stated in the problem.

Thus, the number of sides of the polygon is \( \boxed{12} \).