Polygon
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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 Edit
Polygons are named according to the number of sides, combining a Greekderived numerical prefix with the suffix gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, e.g. 17gon. A variable can even be used, usually ngon. This is useful if the number of sides is used in a formula.
Name  Sides 

Triangle (or trigon)  3 
Quadrilateral (or tetragon)  4 
Pentagon  5 
Hexagon (or sexagon)  6 
Heptagon (avoid "septagon" = Latin [sept] + Greek)  7 
Octagon  8 
Nonagon (or Enneagon)  9 
Decagon  10 
Hendecagon (avoid "undecagon" = Latin [un] + Greek)  11 
Dodecagon (avoid "duodecagon" = Latin [duo] + Greek)  12 
Tridecagon or Triskaidecagon (MathWorld)  13 
Tetradecagon or Tetrakaidecagon (MathWorld)  14 
Pentadecagon (or quindecagon) or Pentakaidecagon  15 
Hexadecagon or Hexakaidecagon  16 
Heptadecagon or Heptakaidecagon  17 
Octadecagon or Octakaidecagon  18 
Enneadecagon or Enneakaidecagon or Nonadecagon  19 
Icosagon  20 
Triacontagon  30 
Tetracontagon  40 
Pentacontagon  50 
Hexacontagon (MathWorld)  60 
Heptacontagon  70 
Octacontagon  80 
Nonacontagon  90 
Hectagon (also hectogon) (avoid "centagon" = Latin [cent] + Greek)  100 
Chiliagon  1000 
Myriagon  10,000 
Naming polygons Edit
To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows
Tens  and  Ones  final suffix  

kai  1  hena  gon  
20  icosa  2  di  
30  triaconta  3  tri  
40  tetraconta  4  tetra  
50  pentaconta  5  penta  
60  hexaconta  6  hexa  
70  heptaconta  7  hepta  
80  octaconta  8  octa  
90  enneaconta  9  ennea 
That is, a 42sided figure would be named as follows:
Tens  and  Ones  final suffix  full polygon name 

tetraconta  kai  di  gon  tetracontakaidigon 
and a 50sided figure
Tens  and  Ones  final suffix  full polygon name 

pentaconta  gon  pentacontagon 
But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17gons and 257gons).
Taxonomic classification Edit
The taxonomic classification of polygons is illustrated by the following graph:
Polygon
/ \
Simple Complex
/ \ /
Convex Concave /
/ \ / /
Cyclic Equilateral
\ /
Regular
 A polygon is called simple if it is described by a single, nonintersecting 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.
The regular polygons most commonly found include:
Somewhat less common are:
See also: tilings of regular polygons
Properties Edit
We will assume Euclidean geometry throughout.
An ngon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for overall size, so 2n4 for shape.
In the case of a line of symmetry the latter reduces to n2.
Let k≥2. For an nkgon with kfold rotational symmetry (C_{k}), there are 2n2 degrees of freedom for the shape. With additional mirrorimage symmetry (D_{k}) there are n1 degrees of freedom.
Angles Edit
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 ngon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular ngon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) turns). This can be seen in two different ways:
 Moving around a simple ngon (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 ngon can be considered to be made up of (n−2) triangles, each of which has an angle sum of π radians or 180°.
Moving around an ngon 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 Edit
Several formulas give the area of a regular polygon:
 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 (x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{n}, y_{n}) of its vertices, listed in order as the area is circulated in counterclockwise fashion, are known. The formula is
 A = ½ · (x_{1}y_{2} − x_{2}y_{1} + x_{2}y_{3} − x_{3}y_{2} + ... + x_{n}y_{1} − x_{1}y_{n})
 = ½ · (x_{1}(y_{2} − y_{n}) + x_{2}(y_{3} − y_{1}) + x_{3}(y_{4} − y_{2}) + ... + x_{n}(y_{1} − y_{n−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 equallyspaced 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 BolyaiGerwien theorem.
Construction Edit
All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).
A regular nsided 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 Edit
In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x_{0},y_{0}) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.
Special casesEdit
Some special cases are:
 Angle of 0° or 180° (degenerate case)
 Two nonadjacent sides are on the same line
 Equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 3132)
 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.
External linksEdit
 Mathworld: Polygon
 Draw n Polygons Applet to draw a polygon with n vertices.