The extended Pi number
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The extended Pi number
Squaring de circle with ruler and compass

The extended Pi is obtained by composition of a pair of internal segments to the circumference, of Pi/2 each, which, as seen in the drawing, are repositioned together forming a capital A, starting with the central segment placed on the diameter and center of the circumference.
The other two go from the upper point (O in the drawing) of the axis and of the circumference, passing united by the lateral ends of the central segment, and ending in their cut with the circumference (point S of the drawing).
Each of these segments represents an approximate value of Pi/2 (0.0009).
In any case, the important thing here is to consider that the extended Pi is a property and element of all the circumferences, and that it can be used for example, to square the circle within the measurement margins that the ruler and compass can do.

Squaring de circle with ruler and compass

From the arithmetic point of view, squaring the circle is not possible because the number pi necessary is irrational, and even an exact division by 2 cannot be done to find the necessary side of the square.
However, and since what is required is to do it with a ruler and compass, in this case it is possible to do it since these devices cannot exceed three decimal digits of precision (aPi-> 0.0009).

Well, the example of squaring the circle that I expose is within these limits of precision of the ruler and compass does not exceed.

As we see in the drawings, this process is done in two steps.
Drawing 1.- Find the approximate segment of Pi,
Drawing 2.- Find the approximate segment on the side of the square.

Approximate and easy squaring by mean of the Phi number.