Question:medium

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

Updated On: Jan 13, 2026
  • 30
  • 54
  • 64
  • None of Above
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The Correct Option is B

Solution and Explanation

The solution begins with established formulas for an \( n \)-sided polygon:

  • Sum of interior angles = \( (n - 2) \times 180 \)
  • Sum of exterior angles = \( 360^\circ \)

The sum of interior angles can also be represented as: \[ (2n - 4) \times 90 \] This is equivalent to \( (n - 2) \times 180 \), as shown by: \[ (n - 2) \times 180 = (2n - 4) \times 90 \]

The problem states the difference between the sum of interior angles and exterior angles is: \[ (2n - 4) \times 90 - 360 = 120n \]

Simplifying the left side yields: \[ (2n - 4) \times 90 - 360 = 180n - 360 - 360 = 180n - 720 \] The equation simplifies to: \[ 180n - 720 = 120n \] Subtracting \( 120n \) from both sides results in: \[ 60n - 720 = 0 \] \[ 60n = 720 \] \[ n = 12 \]

With \( n = 12 \) sides determined, the number of diagonals can be calculated.

The formula for the number of diagonals in an \( n \)-sided polygon is: \[ \frac{n(n - 3)}{2} \] Substituting \( n = 12 \): \[ \frac{12 \times (12 - 3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54 \]

The polygon has \( \boxed{54} \) diagonals.

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