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Calculate p with pencil and paper |  |
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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 |
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 |
| 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.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 |
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 | = |
|
C S = 1 − M S
| C S | = |
|
A C 2 = A S 2 + C S 2
| A C 2 | = |
|
| A C 2 | = |
|
| A C | = |
|
p approximation
look at fig.5 below:
| fig.5 |
-
AB = x0
AC = x1
AD = x2
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.
| xj | = |
|
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 |
| yn |
| xn |
| M T |
| M S |
| M S | = |
|
so
| = |
|
| yn | = |
|
Click the buttons below to watch the step by step approximation of p
The circle has a radius of 1.
