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In fact, it can be said that these calculations are structured as oriented to the quar­ter of a circle, referring to the radius; something that should emerge spon­ta­ne­ous­ly from the report already detected: √Φ × 4; a more enlightening per­spec­tive than taking into account the value of π at 360 ° – the view initialized by the mul­ti­ple polygonal path's approach – and let's see why, this time in a more math­e­mat­i­cal way.
To calculate the length of ¼ of the circumference, reflecting the properties of the square, just geometrically contrast the arc BD with the sum of 2 rays, CB and CD which define it, as base (the root) and height of the arch.
By applying to them, i.e. graphically scaling the perimeter of the virtual square CBED from the vertex in C, the proportion √Φ, we will have obtained the new square Cbed, which intersects the two major sides (or radii: ×2r = 2×r) defining with the sum of the new sides bC and Cd exactly the arc length BD.
BD = 2r × 2√Φ
BD = BA × 		CBED = r²
  CBD = r² ×2√Φ
     or (r ×4√Φ)2
To calculate the area of the quarter circle, since it is2√π the side of the square hav­ing the area of the entire circle, with the same procedure it could be scaled the area of the single quadrant CBED by the22√Φ (0,88665177931216226…), ob­tain­ing the square whose area is π (radius = 1). I hope I was clear enough…
And here is the expected symbiosis of square and circle, deriving directly from the simplest and most practical formulation. A mystery that never existed.
Unless this word of ‘mystery’ could be reflected by my last geometric dis­cov­ery right about that symbiosis, which you can access since Feb. 15, 2022 from the main page, with its little challenge; an assumption which is expanding on the next two pages added.
r+rr×r
on the diameter AB = 1, that is r×2,
2√Φ cut out the length of the
quarter of circumference BD;		 on the radius BC already raised to 2 for the area BCDE,4√Φ cut out the area of the quarter of circle CBD.
Regarding the two radii that delimit the quarter of a unit, their point of con­ver­gence for the circumference constitutes the center; their extremes the pe­riph­er­al points of divergence, are valid for circle and square: to define all the other points equidistant from that center is precisely the square root of the only co­ef­fi­cient of numeral, spatial and gravitative equilibrium, which is the golden sec­tion.
To say the least, a stimulating as well as brilliant example of semantic cor­re­spond­ence, I would dare to say the more realistic if each term finds its right place, re­cur­ring the incidence of the 4 and the square as well as its square root at each step and subtle scan; even if it cannot be adopted by now, it is material for reflection on the real transcendent character of π.
Developing this conception of π, our earthly unit of measurement is proposed as the quarter of a 'turn', which has the same modular function as a [quarter of] square, but it expresses and links together with φ dynamism and growth; where the model is not for example the whole year, but the single season, each in­ter­ac­tion manifests itself with greater likelihood.