How to find the $\pi$?

Question

We know as we increase the number of sides in regular polygon, after infinite repetition it will give us a circle. So, is there any way to find a function which approaches to value of $\pi$ when we consider $\lim\limits_{n\to \infty}$?

Answer 1

The area of a regular $n$-gon circumscribed by the circle of radius $r$ is $$A_n = \frac{1}{2}nr^2\sin\left(\frac{2\pi}{n}\right).$$ You can get the proof here. Taking limit, we will get circle when $n\to\infty$. We can use this formula for $\pi$, $$\pi=\frac{\text{Area of circle with radius r}}{\text{radius}^2}=\frac{1}{r^2}\cdot \lim_{n\to\infty}A_n=\lim_{n\to \infty}\frac{n}{2}\sin\left(\frac{2\pi}{n}\right).$$

Using this result you can tend to $\pi$ putting bigger and bigger values of $n$ in $\frac{n}{2}\sin\left(\frac{2\pi}{n}\right)$. Here I have plotted $\pi -\frac{x}{2}\sin\left(\frac{2\pi}{x}\right)$, for first few values of $x$ very fast, then slowly approaches $0$, means $\frac{n}{2}\sin\left(\frac{2\pi}{n}\right)\to\pi$ very slowly but, the methods explained in the given link in comments tend $\pi$ much faster than this.