Poly Diameter, Radii, Pi
Regular Polygons are plane geometrical figures such as the square or octagon, having all inside corner angles equal, and all sides equal length. Fig- 1 (Unless mentioned otherwise ALL polygons in these writings are regular polygons. Thus, mostly I will drop the adjective, Regular, herein.) The names of the polygons, with the number of sides, are as follows: 3... equilateral or equiangular triangle 4... square, regular quadrilateral or regular quadrangle 5... pentagon 6... hexagon 7... heptagon 8... octagon 9... nonagon 10... decagon 11... undecagon 12... dodecagon 15... pentadecagon. Any polygons not mentions here are designated by their number of sides or angles.
There might be an argument for a one sided polygon being a Point, with that one side being the outside only, with one 360° angle about the point. Likewise a two sided polygon might be a Straight Line with a top and bottom side equaling two, and two end angles of 180° each. I'll skip this topic here, but you might find it interesting to experiment with, after reading some of my findings. Lastly as mentioned earlier, the Hypothesis I am exploring is: Are regular polygons crude circles, and or is the circle an infinite sided regular polygon?
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I started with a square. Traditionally the Radius of a square is a straight line from one corner to the center point. Also traditionally, a straight line drawn perpendicular from any side midpoint to the center of a polygon is called the Apothem. Fig-2 These definitions I presumed to be devised circa when people were drawing lines in the sand, and it was a 50/50 toss as to which shall be where! (I assumed these labels are the same for all regular polygons.) By my reasoning, all regular polygons have the similarities of having three (3) diameters. Fig-3 There is radius-center-radius, apothem-center-apothem, and radius-center-apothem. The only differences between even and odd sided regular polygons is: the even sided polygons have two (2) diameters that can be draw as straight lines end to end, and the odd sided polygons only have one (1) diameter that can be drawn as a straight line end to end. If a diameter is merely defined as the length of two radii, or two apothems__ all polygons can then have center broken straight line diameters.
So which is the true diameter in Fig-3? I imagined if the square were a crude circle then maybe, formulae would be similar? So, would the ratio of the Perimeter or Circumference of a square divided by the diagonal (two radii), be pi for the square? If it is, the formula of pi times radius squared should give the area of the square when the square pi is used... Well, it did not work! However, if the apothem plus apothem diameter of a square is used for the diameter, and divided into the circumference of any square it gives a ratio of 4. And, guess what, the apothem squared times the square pi ratio of 4 will give the area of the square! I expanded this to the other even number sided polygons... Then by dividing their circumference by the twice apothem diameter to get their individual specific pi ratios, and then doing the circle area formula test... A = ¶rČ for area. This also came out ok. I then worked the math for the odd sided polygons. Lo and behold it works for ALL regular polygons!
I believe the apothem is actually the true radius of all regular polygons, and twice the apothem is the true regular polygon diameter. Fig-4
Thus, for my polygon-math, I officially pronounce the Radius of any regular polygon as a straight line from the center point of said figure, to the perpendicular midpoint of any side. The Diameter is simply two Radii. The distance from any polygon center to a corner point is the Co-radius. The Co-diameter would be the length of two Co-radii. The radius plus co-radius is the Irregular-diameter. Fig-4
The above definitions of radii and diameters also apply to the circle. (The obvious, is: We will never know the exact radius or diameter for a circle.) But, now we can use two radii in any direction to represent the diameter, as well as a line straight across through the center point.
I dropped the traditional definitions of radius, apothem, and diameter for regular polygons and for the circle. I also favor circumference instead of perimeter, mostly just to keep the formulae with similar letter designations.
Now we have the first bit of similar consistency in definitions that work the same for all polygons as well as the circle! C/d = ¶, and A = ¶rČ. I show circle pi ratio as the traditional symbol: ¶ only. I show a particular polygon pi as with a sub-script number corresponding to the number of sides, as for a square: ¶4. For polygon pi values in general, or unknown sides, I use a sub-script as: ¶x.
It is important to remember... all the following polygon math herein, unless designated otherwise, will use the above definitions of radius, diameter, and pi!
Note: To find the Side length of any regular polygon... multiply the diameter by its poly pi to get circumference, and divide by the number of sides. S = d * ¶x /n (S = side, n = number of sides)
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Polygon Pi is simply the ratio of any regular polygon circumference divided by poly diameter as defined above. The formula I use for this is:
¶x = C / 2{0.5S/Tan (180°/T)}
¶x = Poly pi, C = Circumference, S = Side length, T = Total number of sides
(If one (1) is used for the side length, S ___ then C = the total number of sides for each polygon is easily calculated. Also when S = 1, the .5S can be dropped & just use .5)
For general interest, without rounding off the end digits compare:
¶circle = 3.14159265358... ¶1000000 = 3.14159265360... The poly pi ratio for a one million sided regular polygon is pretty close to the circle pi ratio. For a bicycle wheel it would roll quite well.
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Poly Pi Listing
| ¶3 = | 5.196152422706631 or √27 | ¶18= | 3.173885652752369 |
| ¶4 = | 4.0 | ¶19= | 3.170539238051492 |
| ¶5 = | 3.632712640026804 | ¶20= | 3.167688806490726 |
| ¶6 = | 3.464101615137754 or √12 | ¶24= | 3.1596599420975 |
| ¶7 = | 3.3710223316527 | ¶25= | 3.158234461152704 |
| ¶8 = | 3.31370849898476 | ¶36= | 3.149591886933264 |
| ¶9 = | 3.275732108395821 | ¶40= | 3.148068272984738 |
| ¶10= | 3.249196962329063 | ¶48= | 3.146086215131435 |
| ¶11= | 3.229891422322034 | ¶50= | 3.145733362682488 |
| ¶12 = | 3.215390309173472 | ¶64= | 3.144118385245904 |
| ¶13= | 3.204212219415707 | ¶75= | 3.14343135305918 |
| ¶14= | 3.195408641462099 | ¶96= | 3.142714599645368 |
| ¶15 = | 3.188348425050332 | ¶100= | 3.142626604335115 |
| ¶16 = | 3.182597878074528 | ¶500= | 3.141633995944886 |
| ¶17= | 3.177850750835611 | ¶1000= | 3.141602989056156 |
