Regular Polygon

A regular polygon is an -sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral). Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.

The terms equilateral triangle and square refer to the regular 3- and 4-polygons, respectively. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, although the terms generally refer to regular polygons in the absence of specific wording.

A regular -gon is implemented in the Wolfram Language as RegularPolygon[n], or more generally as RegularPolygon[r, n], RegularPolygon[x, y, rspec, n], etc.

The sum of perpendiculars from any point to the sides of a regular polygon of sides is times the apothem.

Let be the side length, be the inradius, and the circumradius of a regular polygon. Then

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The area moments of inertia about axes along an inradius and a circumradius of a regular -gon are given by

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(Roark 1954, p. 70).

If the number of sides is doubled, then

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The area of the first few regular -gon with unit edge lengths are

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The algebraic degrees of these for , 4, ... are 2, 1, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 8, 4, 16, 6, 18, 4, ... (OEIS A089929).

The plot above shows how the areas of the regular -gons with unit inradius (blue) and unit circumradius (red) approach that of a unit disk (i.e., ).

If and are the perimeters of the regular polygons inscribed in and circumscribed around a given circle and and their areas, then

			(24)
			(25)

and

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(Beyer 1987, p. 125).

The sum of interior angles in any -gon is given by radians, or (Zwillinger 1995, p. 270).

The following table gives parameters for the first few regular polygons of unit edge length , where is the interior (vertex) angle, is the exterior angle, is the inradius, is the circumradius, and is the area (Williams 1979, p. 33).

polygon
equilateral triangle
square						1
pentagon
hexagon					1
heptagon
octagon
nonagon
decagon
hendecagon
dodecagon
tridecagon
tetradecagon

Only some of the regular polygons can be built by geometric construction using a compass and straightedge. The numbers of sides for which regular polygons are constructible are those having central angles corresponding to so-called trigonometry angles.

It is possible to construct relatively simple two-dimensional functions that have the symmetry of a regular -gon (i.e., whose level curves are regular -gons). Examples, illustrated above, include

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