page_title: Regular polygon
section_title: Circumradius
hierarchical_section_title: Regular polygon / Regular convex polygons / Circumradius
caption_attribution_description: English: Relevant parameters of a pentagon
context_page_description: In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
context_section_description: 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
  For constructible polygons, algebraic expressions for these relationships exist; see 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.
  The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR² where R is the circumradius.
  The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR² − , where s is the side length and R is the circumradius.
caption: Regular pentagon (n = 5) with side s, circumradius R and apothem a
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