The Non Transcendental, Exact Value of π and the Squaring of the Circle

The value of π, elicited from the Area of a given circle, as the mean proportional between the area of the square with the same perimeter as the given circle, and the area of the circumscribing Square to the given Circle.

Lets think of a Square of side length c with an inscribed Circle of diameter 1, and of a Square with the same perimeter as the Circle of side length b.

We can define now π as having the value of 4b

π=4b

In a next step we find out the areas of the two squares and the circle, as follows:

c^2=1

b^2

and

(π*c^2)/4 = (4b*c^2)/4

(4b*c^2)/4 = b

Having now the values of the Areas

c^2; b and b^2

applying the theorem of Pythagoras, we find out the numerical value of b

(b^2)^2 + (b)^2 = 1

b^4 + b^2 - 1 = 0

b = 0.7861513777574233

and

π=4b

π = 3.1446055110296932

Notice:

In fact, before we know that 1 b and b^2 are part of a progression (b^0; b^1; b^2; b^3; b^4...), where the odd exponent represents the area of the circle, the foregoing exponent the area of the circumscribing square and the following one the area of the square with the same perimeter as the circle, it is arbitrary to choose 1 as the hypotenuse of the right triangle 1; b; b^2, derived from the area of the circumscribing square, c^2. Because, the Diameter=1 of the Circle does not determine the Hypotenuse of the triangle 1. b. and b^2.

The Squaring of the Circle

Knowing now that the circles area is equal to the side of the Square with the same perimeter as the circle, we can easily calculate the side of the Square with the same Area as the given Circle, namely:

b' = √b

b' = 0.88665177931216225898564

The Square EAFG has the same Area as the circle of Diameter c=1