Polygon

 For other use please see Polygon (disambiguation)
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.
Contents 
Names and types
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  6 
heptagon (avoid "septagon")  7 
octagon  8 
enneagon (or "nonagon")  9 
decagon  10 
hendecagon (avoid "undecagon")  11 
dodecagon (avoid "duodecagon")  12 
triskaidecagon  13 
pentadecagon  15 
heptadecagon  17 
enneadecagon  19 
icosagon  20 
triacontagon  30 
pentacontagon  50 
hectagon (avoid "centagon")  100 
chiliagon  1000 
myriagon  10,000 
Naming polygons
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 prefix  

kai  1  hena  gon  
20  icosi  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 prefix  full polygon name 

tetraconta  kai  di  gon  tetracontakaidigon 
and a 50sided figure
Tens  and  Ones  final prefix  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
The taxonomic classification of polygons is illustrated by the following tree:
Polygon
/ \
Simple Complex
/ \
Convex Concave
/
Cyclic
/
Regular
 A polygon is called simple if it is described by a single, nonintersecting 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 convex polygon is called concyclic or cyclic polygon if all the vertices lie on a single circle.
 A cyclic polygon is called regular if all its sides are of equal length and all its angles are equal.
Properties
We will assume Euclidean geometry throughout.
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 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). 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 a negative amount one turns.
 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°.
Area
Apothem_of_hexagon.png
Several formulae give the area of a regular polygon:
 <math>A=\frac{nt^2}{4\tan(180^\circ/n)}<math>
 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
All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).
A regular nsided polygon can be constructed with ruler and compass 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 = (x_{0},y_{0}) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.
See also
 cyclic polygon
 geometric shape
 polyform
 polyhedron
 polytope
 simple polygon
 synthetic geometry
 tiling
 tiling puzzleda:Polygon
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