Almost everyone knows the most basic formula for the perimeter of a circle $2\pi r$. I can prove it as follows, but my question is about a simpler idea to show the proof to the students who do not know limit process.
I usually start with a regular n-polygon inscribed in a circle. With respect to the picture we can say $$\text{Perimeter}= \lim_{n \to \infty} \ n \cdot s(n)=l$$ so
$$l= \lim_{n \to \infty} \ n \cdot s(n)=\\ \lim_{n \to \infty} \ n \cdot r\sqrt{2(1-\cos(\frac{2\pi}{n}))}=\\
\sqrt 2 r\lim_{n \to \infty} \sqrt{\frac{n^2\sin^2(\frac{2\pi}{n})}{(1+\cos(\frac{2\pi}{n}))}}=\\
\sqrt 2 r\lim_{n \to \infty} \sqrt{\frac{4\pi^2}{2}}=\\2\pi r$$
Is there another idea to show the proof?
Thanks in advance for any help