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

Question

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Solution:

Step 1: Let the polygon have n sides

Interior angle = (n−2) × 180 / n

Exterior angle = 360 / n

Given: interior − exterior = 120 →

((n−2) × 180 / n) − (360 / n) = 120

Simplify: (180n − 720)/n = 120 → 180 − 720/n = 120 → 720/n = 60 → n = 12

Step 2: Number of diagonals = n(n−3)/2 = 12 × 9 / 2 = 54

Answer: 54

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