Editing Regular polygon (section)

===Circumradius===
[[Image:PolygonParameters.png|class=skin-invert-image|thumb|left|180px|Regular [[pentagon]] (''n'' = 5) with [[edge (geometry)|side]] ''s'', [[circumradius]] ''R'' and [[apothem]] ''a'']]
<div class="skin-invert-image">{{regular polygon side count graph.svg}}</div>
The [[circumradius]] ''R'' from the center of a regular polygon to one of the vertices is related to the side length ''s'' or to the [[apothem]] ''a'' by
:<math>R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} = \frac{a}{\cos\left(\frac{\pi}{n} \right)} \quad_,\quad a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}</math>

For [[constructible polygon]]s, [[algebraic expression]]s for these relationships exist {{xref|(see {{slink|Bicentric polygon|Regular polygons}})}}.

The sum of the perpendiculars from a regular ''n''-gon's vertices to any line tangent to the circumcircle equals ''n'' times the circumradius.<ref name=Johnson/>{{rp|p. 73}}

The sum of the squared distances from the vertices of a regular ''n''-gon to any point on its circumcircle equals 2''nR''<sup>2</sup> where ''R'' is the circumradius.<ref name=Johnson/>{{rp|p. 73}}

The sum of the squared distances from the midpoints of the sides of a regular ''n''-gon to any point on the circumcircle is 2''nR''<sup>2</sup> − {{sfrac|1|4}}''ns''<sup>2</sup>, where ''s'' is the side length and ''R'' is the circumradius.<ref name=Johnson/>{{rp|p. 73}}
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If <math>d_i</math> are the distances from the vertices of a regular <math>n</math>-gon to any point on its circumcircle, then <ref name= Mamuka />

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