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Area of a Circle (Education for Primary & Secondary Schools)

CHAPTER 22.

Dear pupils, I prefer not to bother you with a series of numbers and instead I want to give you a simple presentation of how to draw the area of any circle without measuring it out the way you were taught in school by dividing a length, but rather without measuring and only with a compass, nor will we use the formula of squared radius times Pi (although the end result is the same), since our way is simpler and shorter. One fourth of Perimeter times Diameter: Product = Area of the Circle. If the established Pi perimeter is 3 diameters and a seventh divided into 4 parts = the side, therefore times diameter (hence the other side), the product of this rectangle is the area of the circle. And its equivalence with the opposite rectangle (as will be shown) is likewise the same area, whereas its root is the area of the square. And what it looks like as a draft will be presented without measurements in the simplest way, and this goes for the magnitudes of any and all circles. All you need to remember is your given radius so as to be able to calculate and verify the accuracy of the drawing and thereby of the usual way (of computing) done up to now. There is no difference. The occurrence of differences was because for centuries many struggled to find a geometrical solution but it remained an enigma; but now you will solve it in about fifteen minutes. However don’t forget that proper preparation is almost always a done deal (and it’s always of use in your life and work). So let’s go step by step – preparation followed by a solution without measurements. I repeat: this applies to any size of circle, either of known or unknown diameter.

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A circle of any diameter divided into six parts (normally standard).

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We will inscribe its star-shaped six-sided polygon because it will help us divided the circle into parts.

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Thus with straight lines we will divide the circle into 12 equal parts, counting the poles that we did not depict with straight lines.

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This allows us to draw (dotted line) a square within the circle, which again enables a straight line division of the surface area into 4 parts.

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All this brings about a draft with the same radius as the circle, and from its four diametrical parts we draw a quatrefoil…

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… that then produces an extensive squared circle the sides of which are the size of the diameter or the doubled radius, respectively. That is the preparation. Now we will pass on to the next preparation.

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Draw a straight line and to it we convey the size of the diameter. Now, as you have learned this diameter should be divided without measuring.

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Therefore, into 14 parts. Take 11 parts and with bisector divide them in half (the same would be if we were to divide with 7, but then we would have to take it and divide it into 5.5 parts – (it’s all the same ratio.)

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Now, from the straight line intersection that separates the circle in half, divide the side panels and link them with lengths. Now we have a rectangle with the area of the circle, and its formula would be d2 divided by 14 times 11.

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We now proceed in the same way on the opposite side to equalize the rectangles.

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I don’t have to tell you that the inner square (the boldfaced sides) is the squared circumference, respectively 44 fourteenths or 22 sevenths, or to put it simply Pi = 3,1428571.

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Between it and the spacious square we divide the diagonals into two parts, link them with lengths and thus we have acquired the formula of the squared circle area: squared diameter divided by 14 times 11, and the resulting root is the square’s edge.

And that’s that. All this corresponds to the old way of calculation. Squared radius times Pi (3.1428571¨). As regards the Pi number, don’t let it confuse you in spite of the fact that as of recently all kinds of fabrications have been appearing, and don’t shorten the remainder, i.e. 3.1428571. This remainder is clear. If you divide number 1 (one whole) with 7, you will get this remainder and it stands for one seventh of a whole, in the same way as any remainder belongs to some recognizable single digit, double-digit times two, etc. But this is irrelevant for this chapter of ours. Now you know and are able to draw one of the enigmas (a millennial one, they say) without measuring. That would be that, but I would gladly go further on, introducing the cube of the sphere and in the same simple way linking it with the way that you have been taught and with what you have not bee taught, with the aim that allows you to both calculate it and draw it, But the essence of all this is a geometrical presentation only with a compass and unmarked straightedge. The link to primary and secondary schools is a supplement – for a right triangle but without measures since the school curriculum calls for “division of length segments without measures”. But in setting the postulates we will use a naturally given simplicity without the need for using the “school system” but only a compass and unmarked straightedge, and the method is based on continuation of the squared surface area of the circle (meaning in another dimension) and the sequel is in the third dimension – the surface of a sphere of a radius or diameter.

CROATIA – RIJEKA 28th November, 2014
Author: Tomo Periša
Web: SLIM
English: S.F. Drenovac

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