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Consider an arbitrary regular polygon of nn sides an example of which is shown above. Join all the vertices to the centre to get nn isosceles triangles.
Let ∠OAB=∠OBA=∠OBC=xo.∠OAB=∠OBA=∠OBC=xo.
⇒∠AOB=yo=180o−2xo.⇒∠AOB=yo=180o−2xo.
A−B−D⇒∠ABD=180o.A−B−D⇒∠ABD=180o.
Further, ∠ABD=∠OBA+∠OBC+∠CBD.∠ABD=∠OBA+∠OBC+∠CBD.
⇒∠OBA+∠OBC+∠CBD=180o.⇒∠OBA+∠OBC+∠CBD=180o.
⇒2xo+∠CBD=180o⇒∠CBD=180o−2xo.⇒2xo+∠CBD=180o⇒∠CBD=180o−2xo.
⇒∠CBD=∠AOB=yo.⇒∠CBD=∠AOB=yo.
Each side of the polygon contributes an angle of yoyo to the centre and the sum of all these angle is equal to 360o.360o.
⇒⇒ Number of sides of the polygon, n=360y.n=360y.
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