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

We have a regular polygon where each interior angle exceeds the exterior angle by 120°. We need to find how many diagonals this polygon has.

Interior angle: The angle inside the polygon at each vertex

Exterior angle: The angle you'd turn through if you walked around the polygon's perimeter

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In any regular polygon with n sides:

Interior angle = $\frac{(n-2) \times 180°}{n}$

Exterior angle = $\frac{360°}{n}$

Interior angles of any polygon sum to $(n-2) \times 180°$, so each angle in a regular polygon is this sum divided by n.

Exterior angles always sum to 360° (one full rotation), so each is $\frac{360°}{n}$.

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Given: Interior angle - Exterior angle = 120°

$\frac{(n-2) \times 180°}{n} - \frac{360°}{n} = 120°$

$\frac{(n-2) \times 180° - 360°}{n} = 120°$

$\frac{180n - 360 - 360}{n} = 120°$

$\frac{180n - 720}{n} = 120°$

$180 - \frac{720}{n} = 120°$

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$180 - \frac{720}{n} = 120$

$\frac{720}{n} = 180 - 120 = 60$

$n = \frac{720}{60} = 12$

So our polygon has 12 sides.

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A diagonal is a line segment connecting any two non-adjacent vertices of a polygon.

From each vertex, you can draw lines to all other vertices. Some of these lines are sides (to adjacent vertices), not diagonals. The rest are diagonals.

Total ways to connect any 2 vertices = ${{}}^{n}C_{2} = \frac{n(n-1)}{2}$

Number of sides = $n$

Number of diagonals = ${{}}^{n}C_{2} - n$

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With $n = 12$:

Number of diagonals = ${{}}^{12}C_{2} - 12$

$= \frac{12 \times 11}{2} - 12$

$= 66 - 12 = 54$

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The polygon has 54 diagonals.

Key insight: This method works for any regular polygon - find the number of sides using the angle relationship, then use the diagonal formula.