regular polygon

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. 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 (effectively a straight line), if the edge length is fixed.

General properties

''These properties apply to all regular polygons, whether convex or star.''

A regular ''n''-sided polygon has rotational symmetry of order ''n''.

All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon.

A regular ''n''-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of ''n'' are distinct Fermat primes. See constructible polygon.

A regular ''n''-sided polygon can be constructed with origami if and only if $n = 2^ 3^ p_1 \cdots p_r$ for some $r \in \mathbb$, where each distinct $p_i$is a Pierpont prime.

Symmetry

The symmetry group of an ''n''-sided regular polygon is dihedral group D_''n'' (of order 2''n''): D₂, D₃, D₄, ... It consists of the rotations in C_''n'', together with reflection symmetry in ''n'' axes that pass through the center. If ''n'' is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If ''n'' is odd then all axes pass through a vertex and the midpoint of the opposite side.

Regular convex polygons

All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.

An ''n''-sided convex regular polygon is denoted by its Schläfli symbol . For ''n'' < 3, we have two degenerate cases:
; Monogon : Degenerate in ordinary space. (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any abstract polygon.)
; Digon ; a "double line segment": Degenerate in ordinary space. (Some authorities do not regard the digon as a true polygon because of this.)

In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc.

Angles

For a regular convex ''n''-gon, each interior angle has a measure of:
: $\frac$ degrees;
: $\frac$ radians; or
: $\frac$ full turns,

and each exterior angle (i.e., supplementary to the interior angle) has a measure of $\tfrac$ degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.

As ''n'' approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964°. As the number of sides increase, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line. For this reason, a circle is not a polygon with an infinite number of sides.

Diagonals

For ''n'' > 2, the number of diagonals is $\tfracn(n - 3)$; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces .

For a regular ''n''-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals ''n''.

Points in the plane

For a regular simple ''n''-gon with circumradius ''R'' and distances ''d_i'' from an arbitrary point in the plane to the vertices, we have

:$\frac\sum_^n d_i^4 + 3R^4 = \left(\frac\sum_^n d_i^2 + R^2\right)^2.$

For higher powers of distances $d_i$ from an arbitrary point in the plane to the vertices of a regular $n$-gon, if

:$S^_=\frac 1n\sum_^n d_i^$,

then

:$S^_ = \left(S^_\right)^m + \sum_^\binom\binomR^\left(S^_ - R^2\right)^k\left(S^_\right)^$,

and

:$S^_ = \left(S^_\right)^m + \sum_^\frac\binom\binom \left(S^_ -\left(S^_\right)^2\right)^k\left(S^_\right)^$,

where $m$ is a positive integer less than $n$.

If $L$ is the distance from an arbitrary point in the plane to the centroid of a regular $n$-gon with circumradius $R$, then

:$\sum_^n d_i^=n\left(\left(R^2+L^2\right)^m+ \sum_^\binom\binomR^L^\left(R^2+L^2\right)^\right)$,
where $m$ = 1, 2, …, $n - 1$.

Interior points

For a regular ''n''-gon, the sum of the perpendicular distances from any interior point to the ''n'' sides is ''n'' times the apothemJohnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the ''n'' = 3 case.

Circumradius

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
:$R = \frac = \frac \quad_,\quad a = \frac$

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 2''nR''² 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 2''nR''² − ''ns''², where ''s'' is the side length and ''R'' is the circumradius.

If $d_i$ are the distances from the vertices of a regular $n$-gon to any point on its circumcircle, then

:$3\left(\sum_^n d_i^2\right)^2 = 2n \sum_^n d_i^4$.

Dissections

Coxeter states that every zonogon (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into $\tbinom$ or parallelograms.
These tilings are contained as subsets of vertices, edges and faces in orthogonal projections ''m''-cubes.
In particular, this is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi.
The list gives the number of solutions for smaller polygons.

Area

The area ''A'' of a convex regular ''n''-sided polygon having side ''s'', circumradius ''R'', apothem ''a'', and perimeter ''p'' is given by
:$A = \tfracnsa = \tfracpa = \tfracns^2\cot\left(\tfrac\right) = na^2\tan\left(\tfrac\right) = \tfracnR^2\sin\left(\tfrac\right)$

For regular polygons with side ''s'' = 1, circumradius ''R'' = 1, or apothem ''a'' = 1, this produces the following table: (Note that since $\cot x \rightarrow 1/x$ as $x \rightarrow 0$, the area when $s = 1$ tends to $n^2/4\pi$ as $n$ grows large.)

Of all ''n''-gons with a given perimeter, the one with the larges