Editing Regular polygon (section)

===Points in the plane===

For a regular simple {{mvar|n}}-gon with [[circumradius]] {{mvar|R}} and distances {{mvar|d<sub>i</sub>}} from an arbitrary point in the plane to the vertices, we have<ref>Park, Poo-Sung. "Regular polytope distances", [[Forum Geometricorum]] 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf</ref>

:<math>\frac{1}{n}\sum_{i=1}^n d_i^4 + 3R^4 = \biggl(\frac{1}{n}\sum_{i=1}^n d_i^2 + R^2\biggr)^2.</math>

For higher powers of distances <math>d_i</math> from an arbitrary point in the plane to the vertices of a regular {{mvar|n}}-gon, if

:<math>S^{(2m)}_{n}=\frac 1n\sum_{i=1}^n d_i^{2m}</math>,

then<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 24 February 2025|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

:<math>S^{(2m)}_{n} = \left(S^{(2)}_{n}\right)^m + \sum_{k=1}^{\left\lfloor m/2\right\rfloor}\binom{m}{2k}\binom{2k}{k}R^{2k}\left(S^{(2)}_{n} - R^2\right)^k\left(S^{(2)}_{n}\right)^{m-2k}</math>,

and

:<math> S^{(2m)}_{n} = \left(S^{(2)}_{n}\right)^m + \sum_{k=1}^{\left\lfloor m/2\right\rfloor}\frac{1}{2^k}\binom{m}{2k}\binom{2k}{k} \left(S^{(4)}_{n} -\left(S^{(2)}_{n}\right)^2\right)^k\left(S^{(2)}_{n}\right)^{m-2k}</math>,

where {{mvar|m}} is a positive integer less than {{mvar|n}}.

If {{mvar|L}} is the distance from an arbitrary point in the plane to the centroid of a regular {{mvar|n}}-gon with circumradius {{mvar|R}}, then<ref name= Mamuka />

:<math>\sum_{i=1}^n d_i^{2m}=n\left(\left(R^2+L^2\right)^m+ \sum_{k=1}^{\left\lfloor m/2\right\rfloor}\binom{m}{2k}\binom{2k}{k}R^{2k}L^{2k}\left(R^2+L^2\right)^{m-2k}\right)</math>,
where <math>m = 1, 2, \dots, n - 1</math>.

====Interior points====
For a regular {{mvar|n}}-gon, the sum of the perpendicular distances from any interior point to the {{mvar|n}} sides is {{mvar|n}} times the [[apothem]]<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).</ref>{{rp|p. 72}} (the apothem being the distance from the center to any side). This is a generalization of [[Viviani's theorem]] for the ''n'' = 3 case.<ref>Pickover, Clifford A, ''The Math Book'', Sterling, 2009: p. 150</ref><ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>