General method for drawing any regular polygon given the measurement of one side 

If you love this geometry, then you might love this higher geometry: mathworld.wolfram.com/QuadratrixofHippias.html

This is an exact method if you already have constructed a line segment with the same length as the side of the regular polygon you want to construct.

Nameci2718, you're just flat-out wrong. I would hope the creator didn't intend for this method to be seen as an always-exact method (one doesn't exist for general regular polygons with compass-and-straightedge only). I would assume that either he knows this is an approximation and didn't think that was worth saying, or he was duped as well. Rndm Dud, while what you've said is true, you're missing the point of what Online is saying. What Online is saying is that if you COULD use "perfect" methods perfectly, without any errors, you would get a perfect regular polygon, but if you likewise could use this method perfectly, without any errors, the resulting polygon would unavoidably be NOT regular. True, the error may be small enough for most practical purposes, and may easily be overshadowed by other factors when doing an actual physical construction - but again, that was not the point.

I'm not entirely sure about that. Your expansion seems to be more along the lines of "No method can be perfect in the physical world" which is completely true. To be fair, I don't know exactly what Online's point was, but I took it to be "Even in a perfect world without physical limitations, the method is flawed." That is, dealing with it in a purely mathematical sense. Assuming that is what he meant, your point about physical limitations, while true in our universe, isn't really relevant for the purely mathematical one. As I'm typing this, I realize we may very well have the same opinions, just different focal points - your focus being on the physical world, mine being on the pure mathematics. I assume you would accept that, say, Euclid's construction of a regular pentagon is "perfect" in the pure mathematical sense, though it's impossible to perfectly replicate it in the physical world?

Probably, yeah, the only difference is what we emphasize.

You're welcome; I agree that too often RU-vid comments can be insulting and childish, and while I have a bad habit of sometimes being too blunt, I do try my best to not descend to quite that level. (I just hope I didn't go too far in my comment directed to Nameci...)

A seven-sided polygon (a heptagon) is in fact able to be constructed. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-cErccoHui9g.html

@@kingfuff402 Its close but not 100%

@Murtaza Nek: That's right. In particular, the actual centers for those polygons would *not* be equally spaced. The constructible regular polygons are those whose side count, n, is a product of distinct Fermat ("Fair-mah" - it's French) primes, and any number of factors of 2 (including none). The known Fermat primes are 3, 5, 17, 257, 65537. [To generate these numbers, start with 2, square each term to get the next, then add 1 to all of the terms.] It's unknown whether there are any more (that are prime), but if there are, the next one is larger than an astronomically huge lower bound, and therefore, of absolutely no practical importance. For that matter, neither is 65537. The n-values up to 100 that are and aren't constructible, are: Yes: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96 No: 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100 But using an accurate ruler and some trig (or a good protractor, for that matter!), you can draw any regular polygon with a given side length. With side length s, the height, h, of the center (h is called the apothem of the regular polygon), is h = s/[2 tan(180º/n)] The method shown in this video is exact only for n=6, which is what it started with, anyway. Fred

Straightedge yes, impossible, but marked rulers do make all other polygons constructible. And marked rulers render primitive straightedges obsolete.

Any.

And, the limit step is 1/pi, so by doing an inversion over a unit circle, pi can be obtained, which is impossible, so it also is an approximation going out to infinity, and will slowly become less and less accurate.

A 7-gon cannot be constructed using a compass and straightedge. If it could only be created from a side, it could be "transported" to a circle using parallelograms.

Thank you for your contribution. This is an approximation method for drawing by hand, not for an accurate mathematical solution.

Which method have you got ? Please share it.

@@ArthurGeometry You are correct, sir, this is an approximation. Actually, it's plenty good enough for many "pencil" drawings. As an example, using your method to inscribe a tridecagon (13 sided polygon) with sides 2 units in length, gives a relative error of about 0.2% larger than the "true" circle should be. This is just fine for dividers that use pencil lead. However, with carefully sharpened machinist's dividers, working on metal, this error will be obvious. Similarly, the error will be quite obvious if using a CAD package with precision set smaller than 0.01 units. I'm sure you know this, but for the benefit of other readers, if machinist's/CAD accuracy is required, then a tiny bit of trigonometry will give a precisely accurate solution. As an example, take the above tridecagon with side 2 units long. The radius of the inscribing circle is 1/sin(180/13) = 8.373136 units. In general, the radius is (s/2)/sin(180/n), where s is the length of each side and n is the number of sides. Best regards, Gottfried

construct yours and post for us to watch.

There is no method with the classical tools - straightedge and compass - that will work for most numbers of sides, n; but with a ruler and some trig calculation (or a protractor), it can be done. As accurately as can be measured with those tools, anyway. With straightedge and compass, the scorecard reads as follows... Constructible: n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ... Not constructible: n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, ... Fred