Calculate p with pencil and paper

Introduction
In formula's about circles and spheres we find the constant p.
When the radius is R, then :
 circumference circle = 2pR area circle = pR2 area sphere = 4pR2 volume sphere = (4/3)pR3
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
 fig.1 fig.2 fig.3
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.
 fig.4
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.

A S =  A B 2

M S =  \ 1 − A S 2

M S =
\1 −
2
æ A B 2
ö
­­
èø

C S = 1 − M S

C S =
1 −
\1 −
2
æ A B 2
ö
­­
èø

A C 2 = A S 2 + C S 2

A C 2 =
2
æ A B 2
ö
­­
èø
+
2
æ
1 −
\1 −
2
æ A B 2
ö
­­
èø
ö
­­
èø

A C 2 =
2
æ A B 2
ö
­­
èø
+ 1
−
2
\1 −
2
æ A B 2
ö
­­
èø
+ 1
−
2
æ A B 2
ö
­­
èø

A C

=

\
2 −
2
\1 −
2
æ A B 2
ö
­­
èø
This is the "half-chord" formula.

p approximation
look at fig.5 below:

 fig.5
We define:
AB = x0
AC = x1
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:
p » 2n.xn for high values of n.
using x0,1,2... the "half-chord" formula reads:

xj

=

\
2 −
2
\1 −
2
æ xi 2
ö
­­
èø
where j = i + 1
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:
 fig.6
Because of similarity of the triangles :
 yn xn
=  M T M S

M S =
\1 −
2
æ xn 2
ö
­­
èø
and M T = 1
so
 yn xn

=

1
\1 −
2
æ xn 2
ö
­­
èø

yn

=

xn
\1 −
2
æ xn 2
ö
­­
èø
Action!
Click the buttons below to watch the step by step approximation of p
The circle has a radius of 1.