Editing Regular polygon (section)

==General properties==
[[File:regular star polygons.svg|class=skin-invert-image|thumb|300px|Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols]]
These properties apply to all regular polygons, whether convex or [[star polygon|star]]:

*A regular ''n''-sided polygon has [[rotational symmetry]] of order ''n''.

*All vertices of a regular polygon lie on a common circle (the [[circumscribed circle]]); i.e., they are concyclic points. That is, a regular polygon is a [[cyclic polygon]].

*Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or [[incircle]] that is tangent to every side at the midpoint. Thus a regular polygon is a [[tangential polygon]].

*A regular ''n''-sided polygon can be constructed with [[compass and straightedge]] if and only if the [[odd number|odd]] [[prime number|prime]] factors of ''n'' are distinct [[Fermat prime]]s. {{xref|(See [[constructible polygon]].)}}

*A regular ''n''-sided polygon can be constructed with [[origami]] if and only if <math>n = 2^{a} 3^{b} p_1 \cdots p_r</math> for some <math>r \in \mathbb{N}</math>, where each distinct <math>p_i</math> is a [[Pierpont prime]].<ref>{{Cite thesis|last=Hwa|first=Young Lee|date=2017|title=Origami-Constructible Numbers|url=https://getd.libs.uga.edu/pdfs/lee_hwa-young_201712_ma.pdf|type=MA thesis |publisher=University of Georgia|pages=55–59}}</ref>

===Symmetry===
The [[symmetry group]] of an ''n''-sided regular polygon is the [[dihedral group]] D<sub>''n''</sub> (of order 2''n''): D<sub>2</sub>, [[Dihedral group of order 6|D<sub>3</sub>]], D<sub>4</sub>, ... It consists of the rotations in C<sub>''n''</sub>, together with [[reflection symmetry]] in ''n'' axes that pass through the center. If ''n'' is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If ''n'' is odd then all axes pass through a vertex and the midpoint of the opposite side.