A regular polygon has each interior angle equal to thrice its exterior angle. The number of diagonals it has…

A regular polygon has each interior angle equal to thrice its exterior angle. The number of diagonals it has is:
  1. $14$
  2. $20$
  3. $27$
  4. $35$

Solution

Let exterior $= x$, interior $= 3x$. Then $4x = 180 \Rightarrow x = 45^{\circ}$. So $n = \dfrac{360}{45} = 8$. Diagonals $= \dfrac{8 \cdot 5}{2} = 20$.

Asked in: IMO

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