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). The formulas below apply to convex regular polygons with integer . 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 inside the regular polygon to the lines containing the sides of a regular polygon of sides is times the apothem.

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

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The area moments of inertia about any two axes through the center in the plane of a regular -gon are equal. In particular, axes along an inradius and a circumradius give

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This follows by integrating over the triangles formed by the center and each side.

When the number of sides is doubled, the side length recurrence equation compares regular polygons with the same circumradius . The area recurrence equation instead compares regular polygons with the same inradius :

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The areas of the first few regular -gons with unit edge lengths are

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where denotes the sixth real root of the polynomial in increasing order. 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).

For regular -gons with unit inradius and unit circumradius, respectively, the areas are

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For , these areas satisfy and approach the area of a unit disk as increases.

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

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and

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(Beyer 1987, p. 125).

The sum of interior angles in any simple polygon with sides 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 angle (at a vertex), is the exterior angle, is the inradius, is the circumradius, and is the area (Williams 1979, p. 33). The final column gives the number of diagonals, . The drawings use a common circumradius to compare shapes; the listed lengths and areas refer to unit side length.

representative	polygon							diagonals
	equilateral triangle							0
	square						1	2
	pentagon							5
	hexagon					1		9
	heptagon							14
	octagon							20
	nonagon							27
	decagon							35
	hendecagon							44
	dodecagon							54
	tridecagon							65
	tetradecagon							77

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 positive level curves are regular -gons). Examples, illustrated above, include

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