Calculate p using pencil and paper

Introduction
In formula's about circles and spheres we find the constant p.
When the radius is R, then :
p is about equal to 3,141592654.....but no number exists that is exactly p.
Therefore, in formula's we rather use p instead of say 3,14...so we can substitute later the number of digits
to achieve the accuracy we want.
p can only be approximated: more computing yields a higher accuracy.

This article describes one of many ways to calcultate p, by using a regular polygon to approximate
the circumference of an arc.

The greek mathematician Archimedes used this method at 250 bC.
Using a regular 96 polygon, he found that p was a number between
22
7
 and
223
71

In 1585 , Metius calculated p in 6 digits.
Vieta used a regular 393216 polygon and in 1579 found 9 digits of p.
Adriaen van Roomen , in 1579 , calculated 16 digits and Ludolf van Ceulen, continuing this work,
approximated (1621) p with 35 digits, using a regular 265 polygon.
Thanks to fast computers and power series, today over a million digits of p are known.
This knowledge has no practical application.

Lambert proved 1761 that p is an unmeasurable number.
Lindemann found in 1882 , that p also is transcendental,
meaning that p cannot be the root of any equation.
This implied, that no method can exist to construct a line (by compass and ruler)
having the same length as the circumference of a circle.

The Method
In fig.1 the circumference is approximated by a square,
in fig.2 by a regular octagon and in fig.3 by a regular 16 - polygon.
Starting with a circle having a radius of 1, half the circumference is exactly p.
The more angles, the better the approximation..
Before we start the real work, some initial considerations.

The "half-chord" formula
A chord is a line with both ends on a circle.
In fig.4 , AB and AC are chords. MA is not.
Starting with (the length of) AB, we calculate the length of chord AC.
Note, that MA = MB = MC = 1.
MC is perpendicular to AB, AS = SB, because of symmetry.
Half the circumference is exactly p.

This is the "half-chord" formula.

p approximation
look at fig.5 below:

We define: AB, the diagonal, also is a chord, having a length of 2.
This is a very bad approximation of p.
A little less worse is 2.AC and again a little better is : 4.AD.

much better: using x0,1,2... the "half-chord" formula reads:

Starting with AB = x0 = 2, we calculate x1, with x1 we find x2,
and with x2, x3 is calculated.
This is an iterative process.

Upper limit
The calculated approximation of p will be too small, because the inscribed polygon was used.
By also considering the escribed polygon, an upper limit of p is established.
See fig.6 below:
Because of similarity of the triangles : Action!
Click the buttons below to watch the step by step approximation of p
The circle has a radius of 1.