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A polygon (from the Greek poly, for "many", and gwnos, for "angle") is a closed planar path composed of a finite number of straight line segments. The term polygon sometimes also describes the interior of the polygon (the open area that this path encloses) or to the union of both.

Names and types

A simple non-convex hexagon

A complex polygon

Polygons are named according to the number of sides, combining a Greek root with the suffix -gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, eg 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

The taxonomic classification of polygons is illustrated by the following tree:

                                      Polygon
                                     /       \
                                 Simple     Complex
                                /     \
                           Convex     Concave
                            /
                       Regular

• A polygon is simple if it is described by a single, non-intersecting boundary; otherwise it is called complex.
• A simple polygon is called convex if it has no internal angles greater than 180° otherwise it is called concave.
• A polygon is called regular if all its sides are of equal length and all its angles are equal.

A concyclic or cyclic polygon is a polygon whose vertices all lie on a single circle.

For example, a square is a regular, cyclic quadrilateral.

Properties

We will assume Euclidean geometry throughout.

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. Any simple n-gon can be considered to be made up of (n-2) triangles. The sum of the inner angles of a simple n-gon is therfore (n-2)π radians (or (n-2)180°), and the inner angle of a regular n-gon is (n-2)π/n radians (or (n-2)180°/n).

All regular polygons are concyclic, as are all triangles (see circumcircle).

The area A of a simple polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is

A = 1/2 · (x₁y₂ - x₂y₁ + x₂y₃ - x₃y₂ + ... + x_ny₁ - x₁y_n)

This same formula can also be used to calculate the signed area of complex polygons: follow the sequence of points and count area to the left of your path positive, to the right negative.

If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem.

The question of which regular polygons can be constructed with ruler and compass alone was settled by Carl Friedrich Gauss when he was 19: A regular n-gon can be constructed with ruler and compass if and only if the odd prime factors of n are distinct prime numbers of the form

${\displaystyle 2^{2^{n}}+1}$

These prime numbers are the Fermat primes; the only known ones are 3, 5, 17, 257 and 65537.

See also: geometric shape, polyhedron, polytope, cyclic polygon.