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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.
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BA = r + r [2r]
BD = (r+r) × π/4
| CBED = r × r [r2]
CBD = (r×r) × π/4
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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 π 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.
Ultimately, we could argue that the radius is pertinent to the tracing of the circumference, while the diameter represents its full in-the-round extension, which is the area of the circle.
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