This work proposes a new mathematical adjustment for the number Pi, (the squaring Pi) consisting of a direct main formula and based on the Pythagorean theorem from basic parameters of the circumference such as its radius or diameter, and using in parallel the sides and diagonals of the squares inscribed and circumscribed to the circumference.
Therefore, obtaining Pi by direct formula of its construction parameters, as is done for any other geometric figure.
To support and try to demonstrate the validity of this number Pi (And prove the inaccuracy of the current algorithmic Pi), this work includes the development and proposal of the Circumscription Theorem, which tells us that in the circumscription of regular geometric figures there is an integration of all the component parameters of these circumscribed figures, to form a new figure composed of those that are circumscribed, and with it the ability to make formulas and measurements of all these parameters in function of each other is born.

Keywords: Number Pi, diameter, sides, squares, circumferences, diagonals, circumscription.

Introduction

The first question to be elucidated in this work will be the adjustment of the Squaring Pi through the Pythagorean theorem, which although it can be studied from different points of view and aspects, here we treat it from the point of view and adjustment as a convergent sequence for an Pythagorean exponent N.
This adjustment is carried out using the inscribed square to the circumference, although we can also do it with the circumscribed square.
Well, as can be seen in the drawing, to obtain the semi-square (2.Sqrt2, which delimits Pi as the value of the circumference of radius 1), using the Pythagorean theorem and supporting the radius of the circumference = 1, it is necessary to use and add four times (1^2), with which we use a total of N = 8 as the sum of exponents.
Well, this value of N = 8 is where the convergence of the formula shown in Fig.1 occurs.
As we will see later, and in many other studies that I have published, its simplicity, Pythagorean system of adjustment, also being a direct function of the circumference construction parameters and having properties such as the exponential property, as well as comply with the Circumscription Theorem that we will see later, all that justify it as a special number and a prominent candidate to be the true and correct number Pi.

"In the circumscription of regular geometric figures, for all constructive parameter (or side), there will always be mathematical functions that give us the dimensions of these constructed figures and also the measures of any other parameter of construction, and vice versa"
Logically, in successive circumscription among pairs (or more) of figures, also this circumscription theorem is follows, as it is showed in the drawing (successive circumscription of squares and circumferences).

It must be this way, since when one figure is properly circumscribed over other, what we are actually building is a new figure composed of the previous two and that this union makes all its construction parameters totally related to each other, being able to measure them with common formulas for all of them.
If, for example, we inscribe-circumscribe a square inside of an octagon, immediately all the sides of the square can be measured based on the sides of the octagon, and vice versa, and we can therefore develop a connection and adjustment formula among all the parameters of this composite figure.

The accuracy of Pi through the Circumscription theorem

Therefore and relating basic parameters of the circumference and the circumscribed square, there must be a direct and exact function of the parameter Pi (semicircumference) that gives us for example, the diagonal of the circumscribed square, and vice versa.

2 x Sqrt2 = f(Pi) and vice versa, Pi = f(2 x Sqrt2)

Now well, since here we are studying the number Pi, which is represented as a half-circumference parameter of radius 1, we are going to build successive squares and circumscribed circumferences among them (see drawing), to check if this number Pi complies with the theorem above described.
We see that any parameter (or complete figure) of the circumscribed squares is a function of any other interior or constructor parameter, also belonging to the squares.
In this case, the main and simpest function relative to the square between parameters is Sqrt2,
Then, given the first circumscribed square to the circumference unit that is 8, we have that the following squares are:
8 x (Sqrt2)^n .. = 11.313078 ...; 16; 22.627417 ...; 32, etc.
But, not only this function (Sqrt2) helps us to relate circumscribed squares, but also to relate circumscribed circumferences.
For example, 6.283182888 ... x (Sqrt2)^n ... = 8.8857624554 ..; 12.566365776...; 17.7715249 ....25.13273155... etc.

Short summary

Summarizing, both the square root of 2 (2= diameter of the circumference of radius 1) and the number Pi, (semicircumference of radius 1) are bases that with exponential functions of the same (Sqrt2)^n and (f(Pi))^n, they give us all the successively circumscribed circumferences and squares between them.
Also being these base parameters functions one of the other, as it was put upper : Pi = f(2 x Sqrt2), and vice versa, 2 x Sqrt2 = f(Pi)

For example, with the Squaring Pi, as starting or base parameter.
In circumferences:
(Pi^18)/(10^8) = 8.88576245...
(Pi^35)/(10^16) = 25.13273155...
(Pi^52)/(10^24) = 71.08609964 ...
In squares:
(Pi^34)/(10^16) = 8 ........................With Pi algorithmic = 8. 0001047
(Pi^51)/(10^24) = 22.627417....
(Pi^68)/(10^32) = 64 .................. ....With Pi algorithmic = 64.0016754...
(Pi^102)/(10^48) = 512 .................. With Pi algorithmic = 512.0201055....
(Pi^136)/(10^64) = 4096 .................With Pi algorithmic = 4096.214460...
(Pi^170)/(10^80) = 32.768 ..............With Pi algorithmic = 32770.1446202 ....

Below, scheme of powers of Pi that gives us the exact value of the circumscribed squares and circumferences, given with the Squaring Pi, and the deviation that the Algorithmic Pi produces.

Logically these deviations must exist since the obtaining of the algorithmic Pi is done without using parameters of construction of the circumference, but by approximation with numerical series.

The algorithmic Pi cannot be the correct value of Pi since it does not adapt mathematically to the Circumscription Theorem of regular geometric figures, and when this is applied, the algorithmic Pi gradually distances itself from the real value of the circumscribed geometric figures (successive circumscription of squares and circumferences).
Also, it does not follow the geometric logic of geometric figures measurement based on its build parameters like all other regular geometric figures.
In this sense, logic tells us that a principle of correspondence between geometry and its algebraic measurement has to be fulfilled:
"If geometrically there is a structure and DIRECT correspondence between diameter and circumference, a DIRECT algebraic function must also accompany it that gives us one parameter as a function of the other, and vice versa." That is:
Pi = f(d)
d = f(Pi), Where d is the circumference diameter r=1.

Friends, to remove a bit of seriousness and pageantry from the subject, allow me to put a little fable or allegory about the current assessment or consideration of the algorithmic Pi in front of the squaring Pi.

Author complete study: fermancebo.com: Pi direct formula