Polygon

A polygon (literally "many angle", see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices. If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.

Names and types
Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral, and nonagon are exceptions.

Naming polygons
To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows

That is, a 42-sided figure would be named as follows: and a 50-sided figure

But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons).

Taxonomic classification
The taxonomic classification of polygons is illustrated by the following graph:

The regular polygons most commonly found include:
 * A polygon is called simple if it is described by a single, non-intersecting boundary (hence has an inside and an outside); 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 simple polygon is called equilateral if all edges are of the same length. (A 5 or more sided polygon can be concave and equilateral)
 * A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
 * A cyclic and equilateral polygon is called regular; for each number of sides, all regular polygons with the same number of sides are similar.
 * A complex polygon may also be defined as regular if it is cyclic and equilateral. These are called star polygons.


 * Equilateral triangle
 * Square
 * Regular pentagon
 * Regular hexagon
 * Regular octagon

Somewhat less common are:


 * Regular decagon
 * Regular dodecagon

See also: tilings of regular polygons

Properties
We will assume Euclidean geometry throughout.

An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape.

In the case of a line of symmetry the latter reduces to n-2.

Let k&ge;2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n-1 degrees of freedom.

Angles
Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n&minus;2)&pi; radians (or (n&minus;2)180°), and the inner angle of a regular n-gon is (n&minus;2)&pi;/n radians (or (n&minus;2)180°/n, or (n&minus;2)/(2n) turns). This can be seen in two different ways:
 * Moving around a simple n-gon (like a car on a road), the amount one "turns" at a vertex is 180° minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between -1/2 and 1/2 winding.)


 * Any simple n-gon can be considered to be made up of (n&minus;2) triangles, each of which has an angle sum of &pi; radians or 180°.

Moving around an n-gon in general, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a pentagram and 0° for an angular "eight". See also orbit (dynamics).

Area
Several formulas give the area of a regular polygon:
 * $$A=\frac{nt^2}{4\tan(\pi/n)}$$
 * half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side)

The area A of a simple polygon can be computed if the cartesian coordinates (x1, y1), (x2, y2), ..., (xn, yn) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is
 * A = ½ · (x1y2 &minus; x2y1 + x2y3 &minus; x3y2 + ... + xny1 &minus; x1yn)
 * = ½ · (x1(y2 &minus; yn) + x2(y3 &minus; y1) + x3(y4 &minus; y2) + ... + xn(y1 &minus; yn&minus;1))

The formula was described by Meister in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem.

If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.

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.

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

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.

Point in polygon test
In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.

Special cases
Some special cases are:


 * Angle of 0° or 180° (degenerate case)
 * Two non-adjacent sides are on the same line
 * Equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32)
 * Equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32)

A triangle is equilateral iff it is equiangular.

An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "angular eight" with vertices on a rectangle.