The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Regular Polygon

A regular polygon (or "n-gon") is a convex polygon with equal side lengths and angles. Not all of the regular polygons can be constructed by using a compass and straightedge. The simpler regular polygons have special names:1) equilateral triangle, square, pentagon, hexagon, heptagon, octagon, .....

In general the following equations apply to regular polygons of n vertices:

vertex angle   α = (1-2/n)π
exterior angle   β = 2π/n
sum of angles   Σα = (n-2)π
side length   a = 2ritan(π/n)
a = 2rcsin(π/n)
inradius   ri = rccos(π/n)
ri = a/2cot(π/n)
circumradius   rc = a/2csc(π/n)
rc = risec(π/n)
rc2 = ri2 + a2/4
perimeter   p = na
area   A = n/2ari
A = n/4a2cot(π/n)

1 The names of polygons with more than 4 vertices may refer to either regular or non-regular polygons; however, the terms generally refer to regular polygons if not specified otherwise.

Last Update: 2011-01-11