In

_{1}, ''a''_{2}, ..., ''a_{n}'' and the _{1}, ''θ''_{2}, ..., ''θ_{n}'' are known, from:
:$\backslash beginA\; =\; \backslash frac12\; (\; a\_1;\; href="/html/ALL/s/\_2\_\backslash sin(\backslash theta\_1)\_+\_a\_3\_\backslash sin(\backslash theta\_1\_+\_\backslash theta\_2)\_+\_\backslash cdots\_+\_a\_\_\backslash sin(\backslash theta\_1\_+\_\backslash theta\_2\_+\_\backslash cdots\_+\_\backslash theta\_).html"\; ;"title="\_2\; \backslash sin(\backslash theta\_1)\; +\; a\_3\; \backslash sin(\backslash theta\_1\; +\; \backslash theta\_2)\; +\; \backslash cdots\; +\; a\_\; \backslash sin(\backslash theta\_1\; +\; \backslash theta\_2\; +\; \backslash cdots\; +\; \backslash theta\_)">\_2\; \backslash sin(\backslash theta\_1)\; +\; a\_3\; \backslash sin(\backslash theta\_1\; +\; \backslash theta\_2)\; +\; \backslash cdots\; +\; a\_\; \backslash sin(\backslash theta\_1\; +\; \backslash theta\_2\; +\; \backslash cdots\; +\; \backslash theta\_)$
The formula was described by Lopshits in 1963.
If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points,

star polygon
In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncatio ...

), 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.
The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century.
In 1952, Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where each

_{0},''y''_{0}) lies inside a simple polygon given by a sequence of line segments. This is called the point in polygon test.

pdf

with Greek Numerical Prefixes

with interactive animation

How to draw monochrome orthogonal polygons on screens

by Herbert Glarner

comp.graphics.algorithms Frequently Asked Questions

solutions to mathematical problems computing 2D and 3D polygons

compares capabilities, speed and numerical robustness

Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons {{Authority control Polygons, Euclidean plane geometry

geometry
Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...

, a polygon () is a plane figure
Figure may refer to:
General
*A shape, drawing, depiction, or geometric configuration
*Figure (wood), wood appearance
*Figure (music), distinguished from musical motif
*Noise figure, in telecommunication
*Dance figure, an elementary dance pattern
...

that is described by a finite number of straight line segment
250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B''
In geometry, a line segment is a part of a line that is bounded by two distinct end poi ...

s connected to form a closed polygonal chain
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain ''P'' is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments c ...

or ''polygonal circuit''. The solid plane region, the bounding circuit, or the two together, may be called a polygon.
The segments of a polygonal circuit are called its '' edges'' or ''sides'', and the points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear, d ...

is a 3-gon.
A simple polygon
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If th ...

is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygon
In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncatio ...

s and other self-intersecting polygons.
A polygon is a 2-dimensional example of the more general polytope
In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions ''n'' as an ''n''-dimensi ...

in any number of dimensions. There are many more generalizations of polygons defined for different purposes.
Etymology

The word ''polygon'' derives from theGreek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group
*Greek language, a branch of the Indo-European language family
**Proto-Greek language, the assumed last common ancestor of ...

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''.
Classification

Number of sides

Polygons are primarily classified by the number of sides. See the table below.Convexity and non-convexity

Polygons may be characterized by their convexity or type of non-convexity: *Convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope, ...

: 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.
* 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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnny Ma ...

: the boundary of the polygon does not cross itself. All convex polygons are simple.
* Concave
Concave means curving in or hollowed inward, as opposed to convex.
Concave may refer to:
* Concave function, the negative of a convex function
* Concave lens
* Concave mirror
* Concave polygon, a polygon which is not convex
* Concave set
See also ...

: Non-convex and simple. There is at least one interior angle greater than 180°.
* Star-shaped
in the ordinary sense.
Image:Not-star-shaped.svg, An annulus is not a star domain.
In set ''S'' in the Euclidean space">Set (mathematics)">set ''S'' in the Euclidean space R''n'' is called a star domain (or star-convex set, star-shaped set or r ...

: 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.
* 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 polygon
The term ''complex polygon'' can mean two different things:
* In geometry, a polygon in the unitary plane, which has two complex dimensions.
* In computer graphics, a polygon whose boundary is not simple.
Geometry
In geometry, a complex polygon i ...

'' as one which exists in the complex Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

plane consisting of two complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London (UCL). The Faculty, the UCL Faculty of Engineering Sciences and the UCL Faculty of the Built Envirornment (The Bartlett) toget ...

dimensions.
* Star polygon
In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncatio ...

: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.
Equality and symmetry

* Regular: the polygon is both ''isogonal'' and ''isotoxal''. Equivalently, it is both ''cyclic'' and ''equilateral'', or both ''equilateral'' and ''equiangular''. A non-convex regular polygon is called a ''regularstar polygon
In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncatio ...

''.
* Isogonal or vertex-transitive
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

: all corners lie within the same symmetry orbit
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism gr ...

. The polygon is also cyclic and equiangular.
* Isotoxal or edge-transitive: all sides lie within the same symmetry orbit
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism gr ...

. The polygon is also equilateral and tangential.
* Cyclic: all corners lie on a single circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. T ...

, called the circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polygon ...

.
* Equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other ...

: all edges are of the same length. The polygon need not be convex.
* Equiangular: all corner angles are equal.
* Tangential
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More pre ...

: all sides are tangent to an inscribed circle
(I), excircles, excenters (J_A, J_B, J_C), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle.
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the tri ...

.
Miscellaneous

*Rectilinear
Rectilinear means related to a straight line; it may refer to:
* Rectilinear grid, a tessellation of the Euclidean plane
* Rectilinear lens, a photographic lens
* Rectilinear locomotion, a form of animal locomotion
* Rectilinear polygon, a polygon ...

: the polygon's sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.
* Monotone with respect to a given line ''L'': every line orthogonal
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bilin ...

to L intersects the polygon not more than twice.
Properties and formulas

Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's method consists in assuming a small set of intuitively appealing axioms, ...

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
300px, Internal and external angles
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called ...

– The sum of the interior angles of a simple ''n''-gon is radian
The radian, denoted by the symbol \text, is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in ...

s or degrees. This is because any simple ''n''-gon ( having ''n'' sides ) can be considered to be made up of 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 $\backslash left(1-\backslash tfrac\backslash right)\backslash pi$ radians or $180-\backslash tfrac$ degrees. The interior angles of regular star polygon
In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncatio ...

s were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular $\backslash tfrac$-gon (a ''p''-gon with central density ''q''), each interior angle is $\backslash tfrac$ radians or $\backslash tfrac$ degrees.
* Exterior angle
300px, Internal and external angles
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called ...

– The exterior angle is the supplementary angle
In Euclidean geometry, an angle is the figure formed by two rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles are als ...

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
Turn may refer to:
Arts and entertainment
Dance and sports
* Turn (dance and gymnastics), rotation of the body
* Turn (swimming), reversing direction at the end of a pool
* Turn (professional wrestling), a transition between face and heel
* 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
A pentagram (sometimes known as a pentalpha, pentangle, pentacle or star pentagon) is the shape of a five-pointed star polygon.
Pentagrams were used symbolically in ancient Greece and Babylonia, and are used today as a symbol of faith by many W ...

and 0° for an angular "eight" or antiparallelogram
In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but the sides in the longer pair cross each other as in a scissors mechanism. Antipa ...

, where ''d'' is the density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass per unit volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter '' ...

or turning number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the o ...

of the polygon. See also orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system unde ...

.
Area

In this section, the vertices of the polygon under consideration are taken to be $(x\_0,\; y\_0),\; (x\_1,\; y\_1),\; \backslash ldots,\; (x\_,\; y\_)$ in order. For convenience in some formulas, the notation will also be used. If the polygon is non-self-intersecting (that is,simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnny Ma ...

), the signed area
Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...

is
:$A\; =\; \backslash frac\; \backslash sum\_^(\; x\_i\; y\_\; -\; x\_\; y\_i)\; \backslash quad\; \backslash text\; x\_=x\_\; \backslash text\; y\_n=y\_,$
or, using determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero i ...

s
:$16\; A^\; =\; \backslash sum\_^\; \backslash sum\_^\; \backslash begin\; Q\_\; \&\; Q\_\; \backslash \backslash \; Q\_\; \&\; Q\_\; \backslash end\; ,$
where $Q\_$ is the squared distance between $(x\_i,\; y\_i)$ and $(x\_j,\; y\_j).$
The signed area depends on the ordering of the vertices and of the orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building design ...

of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive -axis to the positive -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
of the absolute value function for real numbers
In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is positive, and if is negative (in whi ...

. This is commonly called the shoelace formula
The shoelace formula or shoelace algorithm (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plan ...

or Surveyor's formula.
The area ''A'' of a simple polygon can also be computed if the lengths of the sides, ''a''exterior angle
300px, Internal and external angles
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called ...

s, ''θ''Pick's theorem
In geometry, Pick's theorem provides a formula for the area of a polygon with integer vertex coordinates in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899, and po ...

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
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by ...

$p^2\; >\; 4\backslash pi\; A$ holds.
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. However, if the polygon is cyclic then the sides ''do'' determine the area. 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).
Regular polygons

Many specialized formulas apply to the areas ofregular polygon
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygon ...

s.
The area of a regular polygon is given in terms of the radius ''r'' of its inscribed circle
(I), excircles, excenters (J_A, J_B, J_C), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle.
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the tri ...

and its perimeter ''p'' by
:$A\; =\; \backslash tfrac\; \backslash cdot\; p\; \backslash cdot\; r.$
This radius is also termed its apothem
Apothem of a hexagon
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to ...

and is often represented as ''a''.
The area of a regular ''n''-gon in terms of the radius ''R'' of its circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polygon ...

can be expressed trigonometrically as:
:$A\; =\; R^2\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash sin\; \backslash frac\; =\; R^2\; \backslash cdot\; n\; \backslash cdot\; \backslash sin\; \backslash frac\; \backslash cdot\; \backslash cos\; \backslash frac$
The area of a regular ''n''-gon inscribed in a unit-radius circle, with side ''s'' and interior angle $\backslash alpha,$ can also be expressed trigonometrically as:
:$A\; =\; \backslash frac\backslash cot\; \backslash frac\; =\; \backslash frac\backslash cot\backslash frac\; =\; n\; \backslash cdot\; \backslash sin\; \backslash frac\; \backslash cdot\; \backslash cos\; \backslash frac.$
Self-intersecting

The area of aself-intersecting polygon
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are:
*the crossed quadr ...

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.
* 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.
Centroid

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are :$C\_x\; =\; \backslash frac\; \backslash sum\_^\; (x\_i\; +\; x\_)\; (x\_i\; y\_\; -\; x\_\; y\_i),$ :$C\_y\; =\; \backslash frac\; \backslash sum\_^\; (y\_i\; +\; y\_)\; (x\_i\; y\_\; -\; x\_\; y\_i).$ In these formulas, the signed value of area $A$ must be used. Fortriangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear, d ...

s (), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for . The centroid
In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a ...

of the vertex set of a polygon with vertices has the coordinates
:$c\_x=\backslash frac\; 1n\; \backslash sum\_^x\_i,$
:$c\_y=\backslash frac\; 1n\; \backslash sum\_^y\_i.$
Generalizations

The idea of a polygon has been generalized in various ways. Some of the more important include: * Aspherical polygon
Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting gre ...

is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualis ...

, a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in cartography
Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using maps. Combining science, aesthetics, and technique, cartography builds ...

(map making) and in Wythoff's construction of the uniform polyhedra
A uniform is a type of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, securit ...

.
* A skew polygon
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The ''interior'' surface (or area) of such a polygon is not uniquely defined.
Skew infinite polygons (apeirogons) have ver ...

does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygon
In geometry, a Petrie polygon for a regular polytope of ''n'' dimensions is a skew polygon in which every (''n'' – 1) consecutive sides (but no ''n'') belongs to one of the facets. The Petrie polygon of a regular polygon is the regular ...

s of the regular polytopes are well known examples.
* An apeirogon
In geometry, an apeirogon (from the Greek words "ἄπειρος" ''apeiros'': "infinite, boundless", and "γωνία" ''gonia'': "angle") or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the ...

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
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. ...

is an infinite sequence of sides and angles that do not lie in a flat plane.
* A complex polygon
The term ''complex polygon'' can mean two different things:
* In geometry, a polygon in the unitary plane, which has two complex dimensions.
* In computer graphics, a polygon whose boundary is not simple.
Geometry
In geometry, a complex polygon i ...

is a configuration
Configuration or configurations may refer to:
Computing
* Computer configuration or system configuration
* Configuration file, a software file used to configure the initial settings for a computer program
* Configurator, also known as choice board, ...

analogous to an ordinary polygon, which exists in the complex plane
Image:Complex conjugate picture.svg, Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The distance along the light blue line from the origin to the point ''z'' is the ''modulus'' or ''absolute value'' of ''z''. The ...

of two real
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

and two imaginary
Imaginary may refer to:
* Imaginary (sociology), a concept in sociology
* The Imaginary (psychoanalysis), a concept by Jacques Lacan
* Imaginary number, a concept in mathematics
* Imaginary time, a concept in physics
* Imagination, a mental faculty ...

dimensions.
* An abstract polygon is an algebraic partially ordered set
250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable.
In mathematics, especially order th ...

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
In geometry, a polyhedron (plural polyhedra or polyhedrons) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as ''poly-'' ...

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
In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions ''n'' as an ''n''-dimensi ...

s. (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.)
Naming

The word ''polygon'' comes from Late Latin ''polygōnum'' (a noun), fromGreek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group
*Greek language, a branch of the Indo-European language family
**Proto-Greek language, the assumed last common ancestor of ...

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
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group
*Greek language, a branch of the Indo-European language family
**Proto-Greek language, the assumed last common ancestor of ...

-derived numerical prefix with the suffix ''-gon'', e.g. ''pentagon'', ''dodecagon''. The triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear, d ...

, 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.Mathworld
Exceptions exist for side counts that are more 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
A pentagram (sometimes known as a pentalpha, pentangle, pentacle or star pentagon) is the shape of a five-pointed star polygon.
Pentagrams were used symbolically in ancient Greece and Babylonia, and are used today as a symbol of faith by many W ...

.
Constructing higher names

To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by Johannes Kepler, Kepler, and advocated by John H. Conway for clarity to concatenated prefix numbers in the naming of quasiregular polyhedron, quasiregular polyhedra.History

Polygons have been known since ancient times. Theregular polygon
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygon ...

s were known to the ancient Greeks, with the pentagram
A pentagram (sometimes known as a pentalpha, pentangle, pentacle or star pentagon) is the shape of a five-pointed star polygon.
Pentagrams were used symbolically in ancient Greece and Babylonia, and are used today as a symbol of faith by many W ...

, a non-convex regular polygon (real
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

dimension is accompanied by an imaginary
Imaginary may refer to:
* Imaginary (sociology), a concept in sociology
* The Imaginary (psychoanalysis), a concept by Jacques Lacan
* Imaginary number, a concept in mathematics
* Imaginary time, a concept in physics
* Imagination, a mental faculty ...

one, to create complex polytope, complex polygons.
In nature

Polygons appear in rock formations, most commonly as the flat facets of crystals, 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 bees is an array of hexagons, and the sides and base of each cell are also polygons.Computer graphics

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. Any surface is modelled as a tessellation called polygon mesh. If a square mesh has 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 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 ''P'' = (''x''See also

* 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 puzzleReferences

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. 461–488.''Notes

External links

*with Greek Numerical Prefixes

with interactive animation

How to draw monochrome orthogonal polygons on screens

by Herbert Glarner

comp.graphics.algorithms Frequently Asked Questions

solutions to mathematical problems computing 2D and 3D polygons

compares capabilities, speed and numerical robustness

Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons {{Authority control Polygons, Euclidean plane geometry