Regular polygon approximation of $\pi$

[elim]

Let $A_n$ be the area of the inscribed regular polygon of unit circle (radius $=1$) with $n$ edges
while $P_n$ be the perimeter of the inscribed regular polygon of unit-dimeter circle.

The Fig below shows the beautiful relation $A_{2n} = P_n$, and the relation between $d_{2n}$ and $d_n$

   

This constitude the elegant procedure in $\pi$ approximation by ancient Chinese mathematician