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 \( \boxed{12} \).