A regular shape has borders that have same length of distance from the center to the sides or in other words with a regular length distribution in it like a circle or a sphere.

In fact, there is no circle or sphere in the reality and all of circles or spheres we can see are regular polygons that contained with triangles or pyramids.

Regular rule is about `pi` number and its growth in regular polygons and regular polyhedrons. In the reality there is not any zero size (sides of polygon or polyhedron). So this rule is used to prove that there is no any infinitive number and obscure number for `pi`. Trough the smallest size, the biggest and accurate numbers can be happen.

In Euclidean geometry, a regular polygon is a polygon that is equilateral (all sides have the same length). The first of them is regular triangle with 3 sides with same length. In this part `n` is the count of sides and `S` is the area of any polygon.

The area of a triangle is `1/2 r a` and `a` as base and `r` as height of the triangle.

Regular polygons are contained with triangles and their area can be calculated by counting triangles area. For example, there are 4 triangles in a square with `theta` insider angle.

`S = 4(1/2 r a) = 2 r a`

`theta = 90`, `alpha = 90`

`a = 2 r`

`S = a^2`

Or the area of a hexagonal:

`S = 6(1/2 r a) = 3 r a`

`theta = 60`, `alpha = 120`

`tan(pi/6) = a/(2r) `

`rarr r = sqrt{3}/2 a`

`S = (3 sqrt{3} a^2)/2`

For `n` sided polygons:

`S = (nra)/2`

`theta = (2pi)/n`

`alpha = pi - theta`

`tan((pi)/n) = a / (2r)`

`rarr a = 2r tan((pi)/n)`

With adding triangles to the polygon it's going to be a circle. So that we have:

`S = lim_{n->∞} (nra)/2 `

`S = lim_{n->∞} nr^2 tan((pi)/n) = pi r^2`

By how many triangles we can put in the polygon `pi` number is going to be accurate.

`lim_{n->∞} (na)/2r = lim_{n->∞} n tan((pi)/n) = pi`

In every regular polygon with n sides there is a `pi` number that i name it `pi_{n}`. For example `pi` of a square is 4 and area of a square is `S = 4 r^2`.

`S = pi_{n} r^2`, `pi_{n} = n tan((pi)/n)`

`n` | `n tan((pi)/n)` | `pi_{n}` |

`3` | `3 tan(pi/3)` | `5.1961524227066`... |

`4` | `4 tan(pi/4)` | `4.0000000000000`... |

`6` | `6 tan(pi/6)` | `3.4641016151378`... |

`8` | `8 tan(pi/8)` | `3.3137084989848`... |

`2^10` | `2^10 tan(pi/2^10)` | `3.1416025102568`... |

`2^100` | `2^100 tan(pi/2^100)` | `3.1415926535898`... |

So in circle there is a accurate pi number, but not enough number because of the smallest particle we can put as base of triangles. a `pi` number is consist of 3 parts. Integer part, correct part and incorrect part. for example in 1024-gonal `pi` number is `3.1416025102568`, `3` is Integer part, `14` is correct part and `16025102568` is incorrect part.

How about `0, 1, 2` for `n`?

In this article regular polyhedrons are made with polygons or in other words `{n, 3}` polyhedrons that n is the side of polygons and 3 is the count of angles in a apex(vertex).

Regular polyhedrons in this article are contained with pyramids and their volume can b calculated with counting pyramids in a regular polyhedron.

This document has not been completed yet and is being edited.

Regular rule - 2018 Apr 24