Editing Polygon

{{Short description|Plane figure bounded by line segments}}
{{Other uses}}
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[[File:Assorted polygons.svg|thumb|400px|right|Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.]]
In [[geometry]], a '''polygon''' ({{IPAc-en|ˈ|p|ɒ|l|ɪ|ɡ|ɒ|n}}) is a [[plane (mathematics)|plane]] [[Shape|figure]] made up of [[line segment]]s connected to form a [[closed polygonal chain]].

The segments of a closed polygonal chain are called its ''[[edge (geometry)|edges]]'' or ''sides''. The points where two edges meet are the polygon's ''[[Vertex (geometry)|vertices]]'' or ''corners''. An '''''n''-gon''' is a polygon with ''n'' sides; for example, a [[triangle]] is a 3-gon.

A [[simple polygon]] is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a '''''polygonal region''''' or '''''polygonal area'''''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. 

A polygonal chain may cross over itself, creating [[star polygon]]s and other [[list of self-intersecting polygons|self-intersecting polygons]]. Some sources also consider closed polygonal chains in [[Euclidean space]] to be a type of polygon (a [[skew polygon]]), even when the chain does not lie in a single plane.

A polygon is a 2-dimensional example of the more general [[polytope]] in any number of dimensions. There are many more [[#Generalizations|generalizations of polygons]] defined for different purposes.

== Etymology ==
The word ''polygon'' derives from the [[Greek language|Greek]] adjective πολύς (''polús'') 'much', 'many' and γωνία (''gōnía'') 'corner' or 'angle'. It has been suggested that γόνυ (''gónu'') 'knee' may be the origin of ''gon''.<ref>{{cite book|title=A new universal etymological technological, and pronouncing dictionary of the English language |first1=John |last1=Craig |publisher=Oxford University |year=1849 |page=404 |url=https://books.google.com/books?id=t1SS5S9IBqUC}} [https://books.google.com/books?id=t1SS5S9IBqUC&pg=PA404 Extract of p. 404]</ref>

==Classification==
[[File:Polygon types.svg|thumb|right|300px|Some different types of polygon]]

===Number of sides===
Polygons are primarily classified by the number of sides.

===Convexity and intersection===
Polygons may be characterized by their convexity or type of non-convexity:
* [[convex polygon|Convex]]: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean.<ref>{{citation |last=Magnus |first=Wilhelm |author-link=Wilhelm Magnus |title=Noneuclidean tesselations and their groups |series=Pure and Applied Mathematics |volume=61 |publisher=Academic Press |year=1974|url=
https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/61/suppl/C|page=37}}</ref>
* Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon.
* [[simple polygon|Simple]]: the boundary of the polygon does not cross itself. All convex polygons are simple.
* [[Concave polygon|Concave]]: Non-convex and simple. There is at least one interior angle greater than 180°.
* [[Star-shaped polygon|Star-shaped]]: the whole interior is visible from at least one point, without crossing any edge. The polygon must be simple, and may be convex or concave. All convex polygons are star-shaped.
* [[list of self-intersecting polygons|Self-intersecting]]: the boundary of the polygon crosses itself. The term ''complex'' is sometimes used in contrast to ''simple'', but this usage risks confusion with the idea of a ''[[Complex polytope|complex polygon]]'' as one which exists in the complex [[Hilbert space|Hilbert]] plane consisting of two [[complex number|complex]] dimensions.
* [[Star polygon]]: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.

===Equality and symmetry===
* [[Equiangular polygon|Equiangular]]: all corner angles are equal.
* [[Equilateral polygon|Equilateral]]: all edges are of the same length.
* [[Regular polygon|Regular]]: both equilateral and equiangular.
* [[Cyclic polygon|Cyclic]]: all corners lie on a single [[circle]], called the [[circumcircle]].
* [[Tangential polygon|Tangential]]: all sides are tangent to an [[inscribed circle]].
* Isogonal or [[vertex-transitive]]: all corners lie within the same [[symmetry orbit]]. The polygon is also cyclic and equiangular.
* Isotoxal or [[edge-transitive]]: all sides lie within the same [[symmetry orbit]]. The polygon is also equilateral and tangential.

The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a ''regular [[star polygon]]''.

===Miscellaneous===
* [[Rectilinear polygon|Rectilinear]]: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
* [[Monotone polygon|Monotone]] with respect to a given line ''L'': every line [[Orthogonal (geometry)|orthogonal]] to L intersects the polygon not more than twice.

==Properties and formulas==
[[File:Winkelsumme-polygon.svg|thumb|upright=1.0|Partitioning an ''n''-gon into {{nowrap|''n'' − 2}} triangles]]
[[Euclidean geometry]] is assumed throughout.

===Angles===
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:
* '''[[Interior angle]]''' – The sum of the interior angles of a simple ''n''-gon is {{nowrap|(''n'' − 2) × [[Pi|π]]}} [[radian]]s or {{nowrap|(''n'' − 2) × 180}} [[degree (angle)|degrees]]. This is because any simple ''n''-gon ( having ''n'' sides ) can be considered to be made up of {{nowrap|(''n'' − 2)}} triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular ''n''-gon is <math>\left(1-\tfrac{2}{n}\right)\pi</math> radians or <math>180-\tfrac{360}{n}</math> degrees. The interior angles of regular [[star polygon]]s were first studied by Poinsot, in the same paper in which he describes the four [[Kepler–Poinsot polyhedron|regular star polyhedra]]: for a regular <math>\tfrac{p}{q}</math>-gon (a ''p''-gon with central density ''q''), each interior angle is <math>\tfrac{\pi(p-2q)}{p}</math> radians or <math>\tfrac{180(p-2q)}{p}</math> degrees.<ref>{{cite book |last=Kappraff |first=Jay |title=Beyond measure: a guided tour through nature, myth, and number |publisher=World Scientific |year=2002 |page=258 |isbn= 978-981-02-4702-7 |url=https://books.google.com/books?id=vAfBrK678_kC&q=star+polygon&pg=PA256}}</ref>
* '''[[Exterior angle]]''' – The exterior angle is the [[supplementary angle]] to the interior angle. Tracing around a convex ''n''-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full [[Turn (geometry)|turn]], so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an ''n''-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple ''d'' of 360°, e.g. 720° for a [[pentagram]] and 0° for an angular "eight" or [[antiparallelogram]], where ''d'' is the [[Density (polytope)#Polygons|density]] or [[turning number]] of the polygon.

===Area===
[[File:Polygon vertex labels.svg|thumb|320px|right|Coordinates of a non-convex pentagon]]

In this section, the vertices of the polygon under consideration are taken to be <math>(x_0, y_0), (x_1, y_1), \ldots, (x_{n - 1}, y_{n - 1})</math> in order. For convenience in some formulas, the notation {{math|1=(''x<sub>n</sub>'', ''y<sub>n</sub>'') = (''x''<sub>0</sub>, ''y''<sub>0</sub>)}} will also be used.

====Simple polygons====
{{further|Shoelace formula}}

If the polygon is non-self-intersecting (that is, [[simple polygon|simple]]), the signed [[area (geometry)|area]] is
:<math>A = \frac{1}{2} \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i) \quad \text {where } x_{n}=x_{0} \text{ and } y_n=y_{0}, </math>
or, using [[determinant]]s
:<math>16 A^{2} = \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} \begin{vmatrix} Q_{i,j} & Q_{i,j+1} \\
Q_{i+1,j} & Q_{i+1,j+1} \end{vmatrix} , </math>
where <math> Q_{i,j} </math> is the squared distance between <math>(x_i, y_i)</math> and <math>(x_j, y_j).</math><ref>B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ.
Math. Debrecen 1, 42–50 (1949)</ref><ref>{{cite web
 |url      = http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf
 |title      = Calculating The Area And Centroid Of A Polygon
 |last      = Bourke
 |first      = Paul
 |date      = July 1988
 |access-date      = 6 Feb 2013
 |archive-date      = 16 September 2012
 |archive-url      = https://web.archive.org/web/20120916104133/http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf
 |url-status      = dead
 }}</ref>

The signed area depends on the ordering of the vertices and of the [[orientation (vector space)|orientation]] of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive {{mvar|x}}-axis to the positive {{mvar|y}}-axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in [[absolute value]]. This is commonly called the ''[[shoelace formula]]'' or ''surveyor's formula''.<ref>{{cite journal |author=Bart Braden |title=The Surveyor's Area Formula |journal=The College Mathematics Journal |volume=17 |issue=4 |year=1986 |pages=326–337 |url=http://www.maa.org/pubs/Calc_articles/ma063.pdf|archive-url=https://web.archive.org/web/20121107190918/http://www.maa.org/pubs/Calc_articles/ma063.pdf|archive-date=2012-11-07 |doi=10.2307/2686282|jstor=2686282 }}</ref>

The area ''A'' of a simple polygon can also be computed if the lengths of the sides, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>'' and the [[exterior angle]]s, ''θ''<sub>1</sub>, ''θ''<sub>2</sub>, ..., ''θ<sub>n</sub>'' are known, from:
:<math>\begin{align}A = \frac12 ( a_1[a_2 \sin(\theta_1) + a_3 \sin(\theta_1 + \theta_2) + \cdots + a_{n-1} \sin(\theta_1 + \theta_2 + \cdots + \theta_{n-2})] \\
{} + a_2[a_3 \sin(\theta_2) + a_4 \sin(\theta_2 + \theta_3) + \cdots + a_{n-1} \sin(\theta_2 + \cdots + \theta_{n-2})] \\
{} + \cdots + a_{n-2}[a_{n-1} \sin(\theta_{n-2})] ). \end{align}</math>
The formula was described by Lopshits in 1963.<ref name="lopshits">{{cite book |title=Computation of areas of oriented figures |author=A.M. Lopshits |publisher=D C Heath and Company: Boston, MA |others=translators: J Massalski and C Mills Jr. |year=1963}}</ref>

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: the former number plus one-half the latter number, minus 1.

In every polygon with perimeter ''p'' and area ''A '', the [[isoperimetric inequality]] <math>p^2 > 4\pi A</math> holds.<ref>{{cite web| url = http://forumgeom.fau.edu/FG2002volume2/FG200215.pdf| title = Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality", ''Forum Mathematicorum'' 2, 2002, 129–130.}}</ref>

For any two simple polygons of equal area, the [[Bolyai–Gerwien theorem]] asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.

The lengths of the sides of a polygon do not in general determine its area.<ref>Robbins, "Polygons inscribed in a circle", ''American Mathematical Monthly'' 102, June–July 1995.</ref> However, if the polygon is simple and cyclic then the sides ''do'' determine the area.<ref>{{cite journal|last=Pak|first=Igor|author-link=Igor Pak|doi=10.1016/j.aam.2004.08.006|issue=4|journal=[[Advances in Applied Mathematics]]|mr=2128993|pages=690–696|title=The area of cyclic polygons: recent progress on Robbins' conjectures|volume=34|year=2005|arxiv=math/0408104|s2cid=6756387}}</ref> Of all ''n''-gons with given side lengths, the one with the largest area is cyclic. Of all ''n''-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).<ref>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref>

====Regular polygons====
Many specialized formulas apply to the areas of [[regular polygon]]s.

The area of a regular polygon is given in terms of the radius ''r'' of its [[inscribed circle]] and its perimeter ''p'' by 
:<math>A = \tfrac{1}{2} \cdot p \cdot r.</math>
This radius is also termed its [[apothem]] and is often represented as ''a''.

The area of a regular ''n''-gon can be expressed in terms of the radius ''R'' of its [[circumscribed circle]] (the unique circle passing through all vertices of the regular ''n''-gon) as follows:<ref>[https://www.mathopenref.com/polygonregularareaderive.html Area of a regular polygon – derivation] from Math Open Reference.</ref><ref>A regular polygon with an infinite number of sides is a circle: <math>\lim_{n \to +\infty} R^2 \cdot \frac{n}{2} \cdot \sin \frac{2\pi}{n} = \pi \cdot R^2</math>.</ref>
:<math>A = R^2 \cdot \frac{n}{2} \cdot \sin \frac{2\pi}{n} = R^2 \cdot n \cdot \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n}</math>

====Self-intersecting====
The area of a [[Complex polygon|self-intersecting polygon]] can be defined in two different ways, giving different answers:
* Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the ''density'' of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure.<ref>{{cite journal|url=http://dynamicmathematicslearning.com/crossed-quad-area.pdf|title=Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral|last=De Villiers|first=Michael|journal=Learning and Teaching Mathematics|volume=2015|issue=18|date=January 2015|pages=23–28}}</ref>
* Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.{{citation needed|date=February 2019}}

===Centroid===
Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are 
:<math>C_x = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i), </math>
:<math>C_y = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i).</math>
In these formulas, the signed value of area <math>A</math> must be used.

For [[triangle]]s ({{math|1=''n'' = 3}}), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for {{math|''n'' > 3}}. The [[centroid]] of the vertex set of a polygon with {{mvar|n}} vertices has the coordinates
:<math>c_x=\frac 1n \sum_{i = 0}^{n - 1}x_i,</math>
:<math>c_y=\frac 1n \sum_{i = 0}^{n - 1}y_i.</math>

==Generalizations==

The idea of a polygon has been generalized in various ways. Some of the more important include:
* A [[spherical polygon]] is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the [[digon]], a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in [[cartography]] (map making) and in [[Wythoff's construction]] of the [[uniform polyhedra]].
* A [[skew polygon]] does not lie in a flat plane, but zigzags in three (or more) dimensions. The [[Petrie polygon]]s of the regular polytopes are well known examples.
* An [[apeirogon]] is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions.
* A [[skew apeirogon]] is an infinite sequence of sides and angles that do not lie in a flat plane.
* A [[polygon with holes]] is an area-connected or multiply-connected planar polygon with one external boundary and one or more interior boundaries (holes).
* A [[Complex polytope|complex polygon]] is a [[configuration (polytope)|configuration]] analogous to an ordinary polygon, which exists in the [[complex plane]] of two [[real number|real]] and two [[imaginary number|imaginary]] dimensions.
* An [[abstract polytope|abstract polygon]] is an algebraic [[partially ordered set]] representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be a ''realization'' of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized.
* A [[polyhedron]] is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are called [[polytope]]s.<ref>Coxeter (3rd Ed 1973)</ref> (In other conventions, the words ''polyhedron'' and ''polytope'' are used in any dimension, with the distinction between the two that a polytope is necessarily bounded.<ref>[[Günter Ziegler]] (1995). "Lectures on Polytopes". Springer ''Graduate Texts in Mathematics'', {{isbn|978-0-387-94365-7}}. p. 4.</ref>)

==Naming==
The word ''polygon'' comes from [[Late Latin]] ''polygōnum'' (a noun), from [[Greek language|Greek]] πολύγωνον (''polygōnon/polugōnon''), noun use of neuter of πολύγωνος (''polygōnos/polugōnos'', the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a [[Greek language|Greek]]-derived [[numerical prefix]] with the suffix ''-gon'', e.g. ''[[pentagon]]'', ''[[dodecagon]]''. The [[triangle]], [[quadrilateral]] and [[nonagon]] are exceptions.

Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.<ref name=mathworld>Mathworld</ref>

Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example the [[regular polygon|regular]] [[star polygon|star]] [[pentagon]] is also known as the [[pentagram]].
{|class="wikitable"
|-
|+ '''Polygon names and miscellaneous properties'''
|-
!style="width:20em;" | Name
!style="width:2em;" | Sides
!style="width:auto;" | Properties
|-
|[[monogon]] || 1 || Not generally recognised as a polygon,<ref>Grunbaum, B.; "Are your polyhedra the same as my polyhedra", ''Discrete and computational geometry: the Goodman-Pollack Festschrift'', Ed. Aronov et al., Springer (2003), p. 464.</ref> although some disciplines such as graph theory sometimes use the term.<ref name="hm96">{{citation
 | last1 = Hass | first1 = Joel
 | last2 = Morgan | first2 = Frank
 | doi = 10.1090/S0002-9939-96-03492-2
 | issue = 12
 | journal = [[Proceedings of the American Mathematical Society]]
 | jstor = 2161556
 | mr = 1343696
 | pages = 3843–3850
 | title = Geodesic nets on the 2-sphere
 | volume = 124
 | date = 1996| doi-access = free}}</ref>
|-
|[[digon]] || 2 || Not generally recognised as a polygon in the Euclidean plane, although it can exist as a [[spherical polygon]].<ref>Coxeter, H.S.M.; ''Regular polytopes'', Dover Edition (1973), p. 4.</ref>
|-
|[[triangle]] (or trigon) || 3 || The simplest polygon which can exist in the Euclidean plane. Can [[triangular tiling|tile]] the plane.
|-
|[[quadrilateral]] (or tetragon) || 4 || The simplest polygon which can cross itself; the simplest polygon which can be concave; the simplest polygon which can be non-cyclic. Can [[square tiling|tile]] the plane.
|-
|[[pentagon]] || 5 || <ref name=namingpolygons/> The simplest polygon which can exist as a regular star. A star pentagon is known as a [[pentagram]] or pentacle.
|-
|[[hexagon]] || 6 || <ref name=namingpolygons/> Can [[hexagonal tiling|tile]] the plane.
|-
|[[heptagon]] (or septagon) || 7 || <ref name=namingpolygons/> The simplest polygon such that the regular form is not [[constructible polygon|constructible]] with [[compass and straightedge]]. However, it can be constructed using a [[neusis construction]].
|-
|[[octagon]] || 8 || <ref name=namingpolygons/>
|-
|[[nonagon]] (or enneagon) || 9 || <ref name=namingpolygons/>"Nonagon" mixes Latin [''novem'' = 9] with Greek; "enneagon" is pure Greek.
|-
|[[decagon]] || 10 || <ref name=namingpolygons/>
|-
|[[hendecagon]] (or undecagon) || 11 || <ref name=namingpolygons/> The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and [[angle trisector]]. However, it can be constructed with neusis.<ref name=Benjamin/>
|-
|[[dodecagon]] (or duodecagon) || 12 || <ref name=namingpolygons/>
|-
|[[tridecagon]] (or triskaidecagon)|| 13 || <ref name=namingpolygons/>
|-
|[[tetradecagon]] (or tetrakaidecagon)|| 14 || <ref name=namingpolygons/>
|-
|[[pentadecagon]] (or pentakaidecagon) || 15 || <ref name=namingpolygons/>
|-
|[[hexadecagon]] (or hexakaidecagon) || 16 || <ref name=namingpolygons/>
|-
|[[heptadecagon]] (or heptakaidecagon)|| 17 || [[Constructible polygon]]<ref name=mathworld/>
|-
|[[octadecagon]] (or octakaidecagon)|| 18 || <ref name=namingpolygons/>
|-
|enneadecagon (or enneakaidecagon)|| 19 || <ref name=namingpolygons/>
|-
|[[icosagon]] || 20 || <ref name=namingpolygons/>
|-
|[[icositrigon]] (or icosikaitrigon) || 23 || The simplest polygon such that the regular form cannot be constructed with [[neusis construction|neusis]].<ref name=Baragar>Arthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151–164, {{doi|10.1080/00029890.2002.11919848}}</ref><ref name=Benjamin>{{cite journal
| last1=Benjamin | first1=Elliot
| last2=Snyder | first2=C
| title=On the construction of the regular hendecagon by marked ruler and compass
| journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
| volume=156
| issue=3
| date=May 2014
| pages=409–424
| doi=10.1017/S0305004113000753| bibcode=2014MPCPS.156..409B
}}</ref>
|-
|[[icositetragon]] (or icosikaitetragon) || 24 || <ref name=namingpolygons/>
|-
|icosipentagon (or icosikaipentagon) || 25 || The simplest polygon such that it is not known if the regular form can be constructed with neusis or not.<ref name=Baragar/><ref name=Benjamin/>
|-
|[[triacontagon]] || 30 || <ref name=namingpolygons/>
|-
|tetracontagon (or tessaracontagon) || 40 || <ref name=namingpolygons/><ref name=Peirce/>
|-
|pentacontagon (or pentecontagon) || 50 || <ref name=namingpolygons/><ref name=Peirce>[https://books.google.com/books?id=wALvAAAAMAAJ&q=hebdomecontagon ''The New Elements of Mathematics: Algebra and Geometry''] by [[Charles Sanders Peirce]] (1976), p.298</ref>
|-
|hexacontagon (or hexecontagon) || 60 || <ref name=namingpolygons/><ref name=Peirce/>
|-
|heptacontagon (or hebdomecontagon) || 70 || <ref name=namingpolygons/><ref name=Peirce/>
|-
|octacontagon (or ogdoëcontagon) || 80 || <ref name=namingpolygons/><ref name=Peirce/>
|-
|enneacontagon (or enenecontagon) || 90 || <ref name=namingpolygons/><ref name=Peirce/>
|-
|hectogon (or hecatontagon)<ref name="drmath"/> || 100 || <ref name=namingpolygons>{{cite book |last=Salomon |first=David |title=The Computer Graphics Manual |url=https://books.google.com/books?id=DX4YstV76c4C&pg=PA90 |date=2011 |publisher=Springer Science & Business Media |isbn=978-0-85729-886-7 |pages=88–90 }}</ref>
|-
| [[257-gon]] <!--please don't add a rarely used English name such as "diacosipentecontaheptagon": it is too long--> || 257 || [[Constructible polygon]]<ref name=mathworld/>
|-
|[[chiliagon]] || 1000 || Philosophers including [[René Descartes]],<ref name=sepkoski>{{cite journal|last=Sepkoski|first=David|title=Nominalism and constructivism in seventeenth-century mathematical philosophy|journal=Historia Mathematica|year=2005|volume=32|pages=33–59|doi=10.1016/j.hm.2003.09.002|doi-access=free}}</ref> [[Immanuel Kant]],<ref>Gottfried Martin (1955), ''Kant's Metaphysics and Theory of Science'', Manchester University Press, [https://books.google.com/books?id=MDe9AAAAIAAJ&pg=PA22 p. 22.]</ref> [[David Hume]],<ref>David Hume, ''The Philosophical Works of David Hume'', Volume 1, Black and Tait, 1826, [https://books.google.com/books?id=4uQBAAAAcAAJ&pg=PA101 p. 101.]</ref> have used the chiliagon as an example in discussions.
|-
|[[myriagon]] || 10,000 || 
|-
| [[65537-gon]]<!--please don't add a rarely used English name such as "hexacismyripentacischilipentacosiatriacontaheptagon": it is too long--> || 65,537 || [[Constructible polygon]]<ref name=mathworld/>
|-
|[[megagon]]<ref>{{cite book |last=Gibilisco |first=Stan |title=Geometry demystified |year=2003 |publisher=McGraw-Hill |location=New York |isbn=978-0-07-141650-4 |edition=Online-Ausg. |url-access=registration |url=https://archive.org/details/geometrydemystif00stan }}</ref><ref name=Darling>Darling, David J., ''[https://books.google.com/books?id=0YiXM-x--4wC&dq=polygon+megagon&pg=PA249 The universal book of mathematics: from Abracadabra to Zeno's paradoxes]'', John Wiley & Sons, 2004. p. 249. {{isbn|0-471-27047-4}}.</ref><ref>Dugopolski, Mark, ''[https://books.google.com/books?id=l8tWAAAAYAAJ College Algebra and Trigonometry]'', 2nd ed, Addison-Wesley, 1999. p. 505. {{isbn|0-201-34712-1}}.</ref> || 1,000,000 || As with René Descartes's example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised.<ref>McCormick, John Francis, ''[https://books.google.com/books?id=KyFHAAAAIAAJ&q=%22million-sided+polygon%22 Scholastic Metaphysics]'', Loyola University Press, 1928, p. 18.</ref><ref>Merrill, John Calhoun and Odell, S. Jack, ''[https://books.google.com/books?id=_aNZAAAAMAAJ&q=%22million-sided+polygon%22 Philosophy and Journalism]'', Longman, 1983, p. 47, {{isbn|0-582-28157-1}}.</ref><ref>Hospers, John, ''[https://books.google.com/books?id=OVu0CORmhL4C&pg=PA56 An Introduction to Philosophical Analysis]'', 4th ed, Routledge, 1997, p. 56, {{isbn|0-415-15792-7}}.</ref><ref>Mandik, Pete, ''[https://books.google.com/books?id=5yHtsM-NToYC&pg=PA26 Key Terms in Philosophy of Mind]'', Continuum International Publishing Group, 2010, p. 26, {{isbn|1-84706-349-7}}.</ref><ref>Kenny, Anthony, ''[https://books.google.com/books?id=ehZGIy_ZYTgC&pg=PA124 The Rise of Modern Philosophy]'', Oxford University Press, 2006, p. 124, {{isbn|0-19-875277-6}}.</ref><ref>Balmes, James, ''[https://books.google.com/books?id=MrwKHqw06hMC&pg=PA27 Fundamental Philosophy, Vol II]'', Sadlier and Co., Boston, 1856, p. 27.</ref><ref>Potter, Vincent G., ''[https://books.google.com/books?id=SnO1FKnJui4C&pg=PA86 On Understanding Understanding: A Philosophy of Knowledge]'', 2nd ed, Fordham University Press, 1993, p. 86, {{isbn|0-8232-1486-9}}.</ref> The megagon is also used as an illustration of the convergence of [[regular polygon]]s to a circle.<ref>Russell, Bertrand, ''[https://books.google.com/books?id=Ey94E3sOMA0C&pg=PA202 History of Western Philosophy]'', reprint edition, Routledge, 2004, p. 202, {{isbn|0-415-32505-6}}.</ref>
|-
|[[apeirogon]] || ∞|| A degenerate polygon of infinitely many sides.
|}

To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows.<ref name=namingpolygons/> The "kai" term applies to 13-gons and higher and was used by [[Johannes Kepler|Kepler]], and advocated by [[John H. Conway]] for clarity of concatenated prefix numbers in the naming of [[quasiregular polyhedron|quasiregular polyhedra]],<ref name=drmath>{{cite web |title=Naming Polygons and Polyhedra |url=http://mathforum.org/dr.math/faq/faq.polygon.names.html |work=Ask Dr. Math |publisher=The Math Forum – Drexel University |access-date=3 May 2015}}</ref> though not all sources use it.
{|class="wikitable" style="vertical-align:center;"
|- style="text-align:center;"
!colspan="2" rowspan="2" | Tens
!''and''
!colspan="2" | Ones
!final suffix
|-
|rowspan="9" | -kai-
|1
| |-hena-
|rowspan=9 | -gon
|-
|20 || icosi- (icosa- when alone) || 2 || -di-
|-
|30 || triaconta- (or triconta-)|| 3 || -tri-
|-
|40 || tetraconta- (or tessaraconta-) || 4 || -tetra-
|-
|50 || pentaconta- (or penteconta-)|| 5 || -penta-
|-
|60 || hexaconta- (or hexeconta-) || 6 || -hexa-
|-
|70 || heptaconta- (or hebdomeconta-)|| 7 || -hepta-
|-
|80 || octaconta- (or ogdoëconta-)|| 8 || -octa-
|-
|90 || enneaconta- (or eneneconta-)|| 9 || -ennea-
|}

==History==
[[File:Fotothek df tg 0003352 Geometrie ^ Dreieck ^ Viereck ^ Vieleck ^ Winkel.jpg|thumb|Historical image of polygons (1699)]]
Polygons have been known since ancient times. The [[regular polygon]]s were known to the ancient Greeks, with the [[pentagram]], a non-convex regular polygon ([[star polygon]]), appearing as early as the 7th century B.C. on a [[krater]] by [[Aristophanes (vase painter)|Aristophanes]], found at [[Caere]] and now in the [[Capitoline Museum]].<ref>{{citation|title=A History of Greek Mathematics, Volume 1|first=Sir Thomas Little|last=Heath|author-link=Thomas Little Heath|publisher=Courier Dover Publications|year=1981|isbn=978-0-486-24073-2|page=162|url=https://books.google.com/books?id=drnY3Vjix3kC&pg=PA162}} Reprint of original 1921 publication with corrected errata. Heath uses the Latinized spelling "Aristophonus" for the vase painter's name.</ref><ref>[http://en.museicapitolini.org/collezioni/percorsi_per_sale/museo_del_palazzo_dei_conservatori/sale_castellani/cratere_con_l_accecamento_di_polifemo_e_battaglia_navale Cratere with the blinding of Polyphemus and a naval battle] {{webarchive|url=https://web.archive.org/web/20131112080845/http://en.museicapitolini.org/collezioni/percorsi_per_sale/museo_del_palazzo_dei_conservatori/sale_castellani/cratere_con_l_accecamento_di_polifemo_e_battaglia_navale |date=2013-11-12 }}, Castellani Halls, Capitoline Museum, accessed 2013-11-11. Two pentagrams are visible near the center of the image,</ref>

The first known systematic study of non-convex polygons in general was made by [[Thomas Bradwardine]] in the 14th century.<ref>Coxeter, H.S.M.; ''Regular Polytopes'', 3rd Edn, Dover (pbk), 1973, p. 114</ref>

In 1952, [[Geoffrey Colin Shephard]] generalized the idea of polygons to the complex plane, where each [[real number|real]] dimension is accompanied by an [[imaginary number|imaginary]] one, to create [[complex polytope|complex polygons]].<ref>Shephard, G.C.; "Regular complex polytopes", ''Proc. London Math. Soc.'' Series 3 Volume 2, 1952, pp 82–97</ref>

==In nature==
[[File:Giants causeway closeup.jpg|thumb|The [[Giant's Causeway]], in [[Northern Ireland]]]]
Polygons appear in rock formations, most commonly as the flat facets of [[crystal]]s, where the angles between the sides depend on the type of mineral from which the crystal is made.

Regular hexagons can occur when the cooling of [[lava]] forms areas of tightly packed columns of [[basalt]], which may be seen at the [[Giant's Causeway]] in [[Northern Ireland]], or at the [[Devil's Postpile]] in [[California]].

In [[biology]], the surface of the wax [[honeycomb]] made by [[bee]]s is an array of [[hexagon]]s, and the sides and base of each cell are also polygons.

==Computer graphics==
{{Main|Polygon (computer graphics)}}
{{more citations needed section|date=October 2018}}
In [[computer graphics]], a polygon is a [[geometric primitive|primitive]] used in modelling and rendering. They are defined in a database, containing arrays of [[vertex (computer graphics)|vertices]] (the coordinates of the [[vertex (geometry)|geometrical vertices]], as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and [[material (computer graphics)|materials]].<ref>{{cite web|url=https://www.khronos.org/opengl/wiki/Vertex_Specification#Primitives|title=opengl vertex specification}}</ref><ref>{{cite web|url=https://msdn.microsoft.com/en-us/library/windows/desktop/bb147325(v=vs.85).aspx|title=direct3d rendering, based on vertices & triangles|date=6 January 2021 }}</ref>

Any surface is modelled as a tessellation called [[polygon mesh]]. If a square mesh has {{nowrap|''n'' + 1}} points (vertices) per side, there are ''n'' squared squares in the mesh, or 2''n'' squared triangles since there are two triangles in a square. There are {{nowrap|(''n'' + 1)<sup>2</sup> / 2(''n''<sup>2</sup>)}} vertices per triangle. Where ''n'' is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).

The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.

In computer graphics and [[computational geometry]], it is often necessary to determine whether a given point <math>P=(x_0,y_0)</math> lies inside a simple polygon given by a sequence of line segments. This is called the [[point in polygon]] test.<ref>{{cite conference|last=Schirra|first=Stefan|editor1-last=Halperin|editor1-first=Dan|editor2-last=Mehlhorn|editor2-first=Kurt|contribution=How Reliable Are Practical Point-in-Polygon Strategies?|doi=10.1007/978-3-540-87744-8_62|pages=744–755|publisher=Springer|series=Lecture Notes in Computer Science|title=Algorithms - ESA 2008: 16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008, Proceedings|volume=5193|year=2008|isbn=978-3-540-87743-1 }}</ref>

==See also==
<!-- Please keep entries in alphabetical order & add a short description [[WP:SEEALSO]] -->
{{div col|colwidth=30em}}
* [[Boolean operations on polygons]]
* [[Complete graph]]
* [[Constructible polygon]]
* [[Cyclic polygon]]
* [[Geometric shape]]
* [[Golygon]]
* [[List of polygons]]
* [[Polyform]]
* [[Polygon soup]]
* [[Polygon triangulation]]
* [[Precision polygon]]
* [[Spirolateral]]
* [[Synthetic geometry]]
* [[Tessellation|Tiling]]
* [[Tiling puzzle]]
{{div col end}}
<!-- please keep entries in alphabetical order -->

==References==

===Bibliography===
* [[Harold Scott MacDonald Coxeter|Coxeter, H.S.M.]]; ''[[Regular Polytopes (book)|Regular Polytopes]]'', Methuen and Co., 1948 (3rd Edition, Dover, 1973).
* Cromwell, P.; ''Polyhedra'', CUP hbk (1997), pbk. (1999).
* Grünbaum, B.; Are your polyhedra the same as my polyhedra? ''Discrete and comput. geom: the Goodman-Pollack festschrift'', ed. Aronov et al. Springer (2003) pp.&nbsp;461–488. ([http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf pdf])

===Notes===
{{Reflist}}

==External links==
{{Wiktionary|polygon}}
{{Commons category|Polygons}}
* {{MathWorld |urlname=Polygon |title=Polygon}}
* [https://web.archive.org/web/20050212114016/http://members.aol.com/Polycell/what.html What Are Polyhedra?], with Greek Numerical Prefixes
* [http://www.mathopenref.com/tocs/polygontoc.html Polygons, types of polygons, and polygon properties], with interactive animation
* [https://web.archive.org/web/20080412002923/http://herbert.gandraxa.com/herbert/dop.asp How to draw monochrome orthogonal polygons on screens], by Herbert Glarner
* [http://www.faqs.org/faqs/graphics/algorithms-faq/ comp.graphics.algorithms Frequently Asked Questions], solutions to mathematical problems computing 2D and 3D polygons
* [https://web.archive.org/web/20110720075903/http://www.complex-a5.ru/polyboolean/comp.html Comparison of the different algorithms for Polygon Boolean operations], compares capabilities, speed and numerical robustness
* [http://dynamicmathematicslearning.com/star_pentagon.html Interior angle sum of polygons: a general formula], Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons

{{Polygons}}
{{Polytopes}}
{{Authority control}}

[[Category:Polygons| ]]
[[Category:Euclidean plane geometry]]