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Straightedge and compass in [medit]action

{the use of the comma for decimals, facilitates the copy / paste on the WIndows™ systems – some whole digits are visible by taping the highlighted areas}
Let's start from scratch, with the two tools at work in the most essential way:

figure 1 is an AB line, or the ruler
figure 2 is a circumference with diameter = AB, or the compass
figure 3 is the combination of the two, namely the ruler and the compass.
When AB = 1,
the circumference is π = 3,14159265358979323846264338327950288419717… to represent the optimal, complete and exhaustive relationship of ruler and com­pass with the transcendence of the π.
BA = r + r   [2r] 
BD = (r+r) × π/4

 
CBED = r × r  [r2]
  CBD = (r×r) × π/4

 

A view under the aspect of the quarters of a circle of diameter 1 already better ex­press­es the relationship between the two formulas for calculating the area of circle and square, based on raising to 2 the 'measure | base side' that occurs in both cases.
For the most correct exhibition of the circumference, is not the & pi; to multiply × 2, but the radius which becomes diameter.

However, since drawing a circle requires a center, doing it starting from the di­am­e­ter requires a graphic operation that divides it into two [fig. 4], centering on A and then on B with radius AB, to obtain the intersection point D from which the central axis perpendicular to AB. Although trivial, it is observed that only the compass pointing to the center can demonstrate the equidistance of op­po­site ends.

All this naturally makes it more practical to define the circle starting from the radius CB, hence the formula universally adopted to measure the cir­cum­fer­ence: π×2r; but in this case the choice will reveal a double raison d'ętre, or rather a quad­ru­ple, useful for the analysis, but destined to return to the di­am­e­ter.