Talk:Regular polygon

Area table

I have reverted the table to this version for the following reasons:

1. The general formula should come first. Ssgxnh moved it back to the bottom. I'd like to insist having first.
2. The reason for the empty cells is WP:UNSOURCED: is it true that "For these polygons the exact area does not simplify out of trig functions"? Perhaps some do simplify, but we just don't know. One of the non-empty cells had the marker as well. I.m.o. an empty cell is really ok. We don't have to put reasons for being empty in there.
3. There is no reason why we should fear the cot-function and replace it with cos/sin. The latter formula does not belong in the approximated values column, not even with parenteses.

Ssgxnh, please propose and discuss on talk page before you make further changes? - Thanks - DVdm (talk) 21:12, 4 August 2010 (UTC)

I have also removed the formula -7 cos(2 Pi/7) / ( 4 sin(2 Pi/7) - 4 ) for the heptagon. It doesn't bring anything new and is more tedious to calculate than the general cot-formula. Furthermore that expression should haven been simplified to 7/4 cos(2 Pi/7) / ( 1 - sin(2 Pi/7) ) and then to 7/4 cot( Pi/7 ), but we already have that in the general formula on top.

Afterall, the idea of the exact values column was to have alternative expressions that do not involve trig functions. DVdm (talk) 07:43, 5 August 2010 (UTC)

The paragraph after the table says: "The amounts that the areas are less than those of circles with the same perimeter, are (rounded) equal to . . . ." As a non-mathematician Math Olympiad coach, I'd love to see a table or graph here showing the increasing area of regular polygons of equal perimeter, approaching but never reaching the area of a circle with the same circumference (last entry could be a circle). The existing chart keeps the side lengths at 1 unit, which gives astronomical areas for higher-number n-gons but doesn't make it easy to directly compare the areas of the various polygons. Thanks. JayBeeEye (talk) 17:54, 1 November 2011 (UTC)

Also, shouldn't the "exact area" for the square be the formula s^2 and the "approximate area" be "1"? Seems like the "1" is in the wrong column, even though it is exact, not approximate. Thanks. JayBeeEye (talk) 18:09, 1 November 2011 (UTC)

 Done. That's a good idea. I have expanded the table with 3 colums for r=1 and ditto for a=1. The calculations are straightforward and were calculated with Maple 13. I guess we can assume that we have a consensus that this conforms to the spirit of wp:CALC. In any case, I have added a reference to the defined MAPLE function. DVdm (talk) 17:00, 4 November 2011 (UTC)

Is the table incomplete? Some of the exact areas in the table aren't listed. Matt.syl (talk) 15:01, 15 May 2018 (UTC)

Some of the exact values are either unknown, or perhaps not expressable in radical form. - DVdm (talk) 15:28, 15 May 2018 (UTC)

Interior angles

The measure of an interior angle of a regular n-gon is an integer number of degrees if and only if n divides 360. There are only 22 such n, namely, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. For those values of n, the measures of the interior angles are 60°, 90°, 108°, 120°, 135°, 140°, 144°, 150°, 156°, 160°, 162°, 165°, 168°, 170°, 171°, 172°, 174°, 175°, 176°, 177°, 178°, and 179° respectively. GeoffreyT2000 (talk) 16:35, 18 April 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Regular polygon/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This article jumps into complexities too quickly. It needs refocussing with a gentler start and a section on History, e.g. geometric construction by the Greeks, constructing the heptagon, Poinsot's study of regular star polygons (in the same paper where he describes the regular star polyhedra). That table of area results should go - all we need is the general method and a couple of worked examples. More pictures too, especially of star polygons. -- Steelpillow 22:37, 8 June 2007 (UTC) PS1234 (talk) 09:14, 14 March 2014 (UTC) === A regular polygon , one side as base of polygon === H = height of regular polygon , W = width of regular polygon , r = circumradius of regular polygon , n= number of sides of regular polygon H = 2r•cos2(Π/2n) if n mod 2 = 1 , 2r•cos(Π/n) if n mod 2 = 0 W = 2r•cos(Π/2n) if n mod 2 = 1 , 2r•cos(Π/n) if n mod 2 = n mod 4 = 0 , 2r if n mod 2 = 0

Last edited at 14:23, 14 March 2014 (UTC). Substituted at 02:33, 5 May 2016 (UTC)

Area formulae

I added \tfrac{p^2}{4n} \cot{\tfrac{\pi}{n}} as it was the form I was looking for when I loaded the page (and none of the others are in terms of just p,n). Is general policy to avoid any redundant formulae? If so I'm not sure why nsa/2 is there as it's also clearly pa/2. @Wcherowi: Indivicivet (talk) 17:21, 10 December 2016 (UTC)

There are literally thousands of ways these formulas can be written, so one needs to make some choices as to which ones should be given. The formula nsa/2 is present because it is the clearest link to the area of a triangle form and so, the most basic version. Using ns = p gives the simplest form using the perimeter. Your addition was a second application of this and I did not see the need for it as it did not add anything novel to the string of formulae.--Bill Cherowitzo (talk) 19:13, 10 December 2016 (UTC)

Origami Constructibility

I recently added a theorem proved in 2004 by Hwa Young Lee about origami constructible polygons (she also proved many other things in the paper; it's worth a read if you haven't). I'm not sure I formatted the citation correctly. Help would be appreciated! DancingGrumpyCat^talk | (ze/zir or she/her) 16:22, 9 January 2022 (UTC)