# Proving that a regular polygon with infinite sides is a circle by using limits on the formula $\frac{\pi}{n}(n-2)$

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.

• 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 Oct 6 '15 at 11:20
• Possible duplicate of math.stackexchange.com/questions/97861/…. – lhf Oct 6 '15 at 11:54
• @lhf, see the edit. – Aditya Agarwal Oct 6 '15 at 13:03
• – Martin Sleziak Oct 6 '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 6 '15 at 13:58

## 1 Answer

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:

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

• Thank you for an approach. But my question is different. This doesn't answer mine. I will edit to explain how – Aditya Agarwal Oct 6 '15 at 13:01
• The interior angle of a polygon is $alpha=\pi(n-2)/n$. So you formula is not correct, and see the add to my answer. – Emilio Novati Oct 6 '15 at 13:11
• What is $\theta$? – Aditya Agarwal Oct 6 '15 at 13:18
• Defined in my answer: the angle at center $C$ that subtends a side of the polygon. – Emilio Novati Oct 6 '15 at 13:19
• If $\theta \to 0$ how it is a circle? – Aditya Agarwal Oct 6 '15 at 13:20