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In childhood, when we were taught circles for the first time, our teacher always told us that a circle is like a polygon which has infinite sides. But how to prove it?
A regular polygon's interior angle is given by $\frac\pi n(n-2)$ and when we use limits, $$\lim\limits_{n\to\infty}\frac\pi n(n-2)=\pi$$ But now how do we use it to prove that this polygon is a circle in fact.
Edit: We have to use this formula to prove that a polygon with infinite size is a circle.

Aditya Agarwal
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    This simply doesn't make sense---a polygon by definition has finitely many sides. One can say that a circle is (in various senses) a limit of a suitable sequence of polygons. – Travis Willse Oct 06 '15 at 11:20
  • Possible duplicate of http://math.stackexchange.com/questions/97861/limit-of-the-sequence-of-regular-n-gons. – lhf Oct 06 '15 at 11:54
  • @lhf, see the edit. – Aditya Agarwal Oct 06 '15 at 13:03
  • Some related posts: http://math.stackexchange.com/questions/31785/how-many-sides-does-a-circle-have, http://math.stackexchange.com/questions/1164977/how-to-prove-the-infinite-number-of-sides-in-a-circle and http://math.stackexchange.com/questions/478005/how-can-i-show-that-a-sequence-of-regular-polygons-with-n-sides-becomes-more-a – Martin Sleziak Oct 06 '15 at 13:53
  • @MartinSleziak, that image question is really really great! Thank you! But my question specifically uses the interior angle formula. – Aditya Agarwal Oct 06 '15 at 13:58

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A proof depends by the definition of a circle that we use.

If the definition is:

the locus of points equidistant from a given point called center $C$

consider a regular polygon with $C$ as center of symmetry. For a point on the polygon the distance $d$ from $C$ is such that: $$ r\le d\le r\cos \dfrac{\theta}{2} $$ where: $r$ is the distance of a vertex form $C$ and $\theta$ is the angle of vertex $C$ subtended by a side.

If the number of sides $n \rightarrow \infty$ than $\theta \rightarrow 0$ and : $$ \lim_{\theta \to 0}r\cos \dfrac{\theta}{2}=r $$ so, at the limit, all points of the polygon have the same distance $r$ from $C$.

if you want use the internal angle $\alpha=\dfrac{\pi}{n}(n-2)$, note that $\theta = \pi -\alpha$ and: $$ n \rightarrow 0 \iff \alpha \rightarrow \pi \iff \theta \rightarrow 0 $$

or use:

$$ r\le d\le r \sin \dfrac {\alpha}{2} $$

Emilio Novati
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