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Computing the magnitude of an arbitrary angle



Unique case – the triangles of the hexagon are equilateral triangles.
The chords of every hexagon segment are equal to the sides of the angle – radius of the full angle.

360° ÷ 6 x 1 = 60°
ili 360° x 1 ÷ 6 = 60°

every next chord divides its circle (arc of the full angle) differently, according to the principle of the hexagonal formula.

In this chapter, before we begin with the construction of the nonagon and its related polygons, we will clarify a „luminosity“ that will enlighten a succession of unknowns in the literature of geometry throughout the history of civilization. We won’t discuss whether these were known or unknown, but instead we will stick to the rules of geometrical reality. This “luminosity” will always be before our eyes whenever we are considering any construction, or any kind of polygon. Moreover, it will offer not only the possibility of their construction but also the hitherto incredible possibility to ascertain the magnitude of any unknown angle in a very simple way (only by using the compass, geometry’s basic device). Today this “luminosity” has been substituted by another device – the protractor, which appears to be easier and more practical. But it is known that in the absence of basic and elementary cognitions we cannot reach the essence of any of the natural sciences; that would be like transferring a grade school child to an educational institution of higher learning. It is in man’s nature to learn step-by-step.

Therefore, with this tool of geometry we stick one of its peaks into a point in space. That point is the midpoint – the beginning. The other end of the compass with its circling around the midpoint limits the space with its arc. This limitation of space with the compass that commenced around a single point that we call the initial point of the circular arc, finishes in that same point. Thus we have limited space with the use of the compass. We have agreed that the circular arc has 360° and that it forms a full circle. Every circle begins with an initial point and finishes there as its endpoint. This may seem banal and so hackneyed to the extent that it may give rise to scorn among grownups. However, let’s repeat once again: since ancient times it has been proven that the span of a compass stuck into a circular arc and moving from point to point of its span, it divides the circular arc (full angle) into six equal sections, i.e. by its going forward from the initial point of the circular arc and finishing there, forming a full circular arc separated into six sections, namely 360° ÷ 6.  We named this span a semi-diameter or a radius. The radius divides its circle into six equal parts. Thus the full angle is divided into six equal parts. What is the magnitude of one part? Of course, we said: 360° ÷ 6 = 60°. That is the keystone for computing the angle. But 360° ÷ 6 = 60° is not right. If we want to get the correct result, we must not dismiss even the most trivial element no matter how banal it seems. Such an omission would disrupt understanding and we should look at such a deletion as shallowness or stubbornness, and that would take us to a host of “impossibilities” in a natural science that does not put up with that. Hence, 360° ÷ 6 = 60° is not the magnitude of one of the six segments of a divided full angle with the radius of the circular arc, but instead the correct formula should be: 360° ÷ 6  x 1 = 60°.  Why is that “times one” so essential?

First of all, let’s establish what we have learned.

  1. The spacing taken by the compass, i.e. the radius that depicts a portion of the plane, limits it with a circular arc and thus makes a full circle.
  2. This same spacing divides the circular arc into six equal parts, hence, divides the full angle into six equal angular parts.
  3. The magnitude of one part of a full angle is found when we divide the full angle with six and multiply it with one .

That is the only case when this is done in such a way. And it is precisely such unique cases in the geometry of schemes, codes, principles or call them as you wish, according to their known laws (patterns) that all other cases of unknowns can be solved. Why is that a unique case?  Each section of a divided circular arc has an additional dimension, i.e. each arc has a chord. In a case like this one the chord of one section is equal to the radius of a full angle. In any other case this no longer occurs. Therefore we have acquired, when we connect the center with lengths of divisional points, lengths that are equal to the radius. In other words, we get six equilateral triangles, thus we can say – we get a hexagon. And what happens when the chord is not the same as the radius? What then is the magnitude of the angle? Bigger or smaller if the chord is bigger or smaller? We will get isosceles triangles (the chord as baseline, the radius as sides). How will we learn the magnitude of the vertex? So what does the guidance of that “unique case” tell us: 360 divided by the radius of the circle’s chord of the arbitrary angle multiplied with the number of partitions emerging on the arc of the arbitrary angle.

Therefore, in this way it is possible to calculate the magnitude of any arbitrary angle. In the case of isosceles triangles all three angles, and in the case of scalene triangles at least two angles; whereas for scalene polygons there are many possibilities – also only with the use of a compass. The pattern that we have learned will prove to be the fundamental element in solving unknowns that have been passed on to us from ancient times (the 7-sided and nine-sided polygon, squaring the circle, doubling the cube, relationships of squaring geometrical figures, etc), unknowns that are in the natural sciences, mathematics, geometry… still marked as “impossible”.  However, I repeat, that there are no unknown or impossibilities in geometry. But if you ask any contemporary mathematician whether it is possible to calculate the magnitude of an arbitrary angle only with a compass, his first answer would be: “impossible”!. True, we have only one datum: the full angle of 360°. But in the first instance, this datum does not come to mind. The next datum is the chord of the angle’s arc. It is taken as the divisional “device” of the compass, hence the only known numeric datum is the full angle of  360°.

Now we come to the conclusion that may be looked upon with scorn:
– every circle begins with an initial point and finishes there as its endpoint, which means that every division  of the circle’s arc of any arbitrary angle begins in one point, the vertex of the chord of the arc and ends there. When we perform the division of the chord, we simply divide 360° with the total number of partitions and multiply with the number of divisions in the segment (arc) of the arbitrary angle and that is how acquire its magnitude. Logical,  simple and clear.

In addition to this, there is a sequence of other oversights in modern geometry (as there are in the whole science of mathematics) so it is no wonder many illogicalnesses have occurred. So it is our assignment to eliminate them in ways that are strictly geometrical, i.e.:

  1. with compass only – by rounded, circular plotting
  2. in straight linea – with unmarked straightedge and circumscribing (with compass)

Thus with a succession of geometrical examples we will “clear up” what has been “blurred”, segment by segment. However, in order to more clearly elucidate the beginnings of our understanding, we will go step-by-step allowing ourselves some diminutions – incomplete plotting, at least at the start of this extraordinary large opus.

* * *


angle less than 60°

An arbitrary angle of unknown magnitude is given. Inscribe the angle’s arc, of arbitrary radius, with its full circle.

* * *

Inscribe the chord and from the vertex angle (center) check out the chord’s radius. We see that its radius is somewhat lesser than the radius of the arc. We take length A-ZB and apply it to the compass and implement the pattern. Every division of a circle begins with its initial point (A-Z) and ends in it.

* * *

360° ÷ 31 = 11,612903 x 5 = 58,064515°

From point A-Z along the radius (magnitude) we divide it all the way until we return to point A-Z. In this case we have a total of 31 partitions (31 isosceles triangles of equal size). The vertex of each of these angles is 360° ÷ 31. We multiply this result with 5 (since the given angle has 5 such isosceles triangles), of the now known magnitudes of vertex angles and thereby we get the magnitude of the given arbitrary angle (which one can check with a protractor).

* * * *


Inscribe the randomly given angle with the arc of the arbitrary radius, and along its chord (A-ZB) A-Z is the starting point and endpoint of the divided chord.

* * *

Here we see that the chord’s radius is larger than the radius of the arc of the arbitrary given angle. In the compass we adopt the size of the chord and start the division on the circle of the arbitrary given angle from point AZ where the division will come to an end.

* * *

360° ÷ 14 = 25,714285 x 3 = 77,142855°

In this example the length of the chord divides the arc (of circle) of given angle into 14 partitions. The division gives us isosceles triangles whose vertex angle is 360° ÷ 14, and since the given angle has three such triangles we multiply it with three to get the magnitude of the arbitrary given angle.
Let’s recapitulate: in this way it is possible, in a strictly geometric manner, to calculate any arbitrary angle – as we have seen in these examples that we plotted in completely abridged patterns like the ones applied in present-day geometry for the sake of making this setting out more clear-cut.
Otherwise, in sacred ancient natural geometry divisions are shown in full circles for a number of reasons (the control of divisional accuracy, the products of division, etc., and the beauty of equilibrium and harmony of every division, hence by those means, the elimination of “esoteric” afterthoughts).
Besides being demonstrated for the first time in our literate civilization, this refreshingly novel approach will serve us in the forthcoming texts (chapters dealing with the 9-sided and 18-sided polygon – with examples of their trisection and other “enigmas”).
That’s why it’s important to remember:

  • every circle begins with an initial point and ends in it, and
  • every division begins with an initial point and ends in that point regardless of how many times this is done.

* * *

360° ÷ 39 = 9,2307692° x 3 = 46, 153846°

In sacred geometry proper plotting of an arbitrary given angle is by means of full circles division of the chord radius magnitude along the arc (of its circle). The reason for this is always because it affords a multiple check of the divisional correctness of a number of other data about which we will say more on some other occasion.

* * *

360° ÷ 40 = 9° x 11 = 99°

As in the example of the arbitrary given angle smaller than 60°, the same procedures goes for the angle in excess of 60°. One hundred percent accuracy will be attained one day when we have computerized simulation programs of calculation for any arbitrary angle by means of division of its chord radius along its arbitrarily determined arc. Perhaps one day!

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2 Responses to “Computing the magnitude of an arbitrary angle”

  1. ivo says:

    Primjmjer B nije toćna formula nije puta 5 nego puta 3

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