In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is -

The correct answer is 54.

Let us solve the problem step-by-step to find the number of diagonals of the regular polygon.

Step 1: Understand the relationship between interior and exterior angles of a regular polygon.
For any regular polygon, the sum of an interior angle and its corresponding exterior angle at any vertex is always equal to 180 degrees. Let the interior angle be denoted by $i$ and the exterior angle be denoted by $e$.
Therefore, we have the first equation:
$i+e=180$

Step 2: Use the given information.
The problem states that the interior angle exceeds the exterior angle by 120 degrees. This gives us the second equation:
$i-e=120$

Step 3: Solve for the exterior angle ($e$).
Subtracting the second equation from the first equation, we get:
$(i+e)-(i-e)=180-120$
$2e=60$
$e=30$
So, the exterior angle of the regular polygon is 30 degrees.

Step 4: Find the number of sides ($n$) of the polygon.
The sum of all exterior angles of any polygon is always 360 degrees. Since it is a regular polygon with $n$ sides, each exterior angle is given by:
$e=\frac{360}{n}$
Substituting $e=30$ into the equation:
$30=\frac{360}{n}$
$n=\frac{360}{30}=12$
Thus, the polygon has 12 sides (it is a regular dodecagon).

Step 5: Calculate the number of diagonals.
The formula for the number of diagonals of a polygon with $n$ sides is given by:
$\text{Number of diagonals}=\frac{n(n-3)}{2}$
Substituting $n=12$ into the formula:
$\text{Number of diagonals}=\frac{12(12-3)}{2}$
$\text{Number of diagonals}=\frac{12\times 9}{2}$
$\text{Number of diagonals}=6\times 9=54$

Therefore, the number of diagonals of this polygon is 54.