Polygon

For other uses, seePolygon (disambiguation).

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.

Ingeometry,apolygon(/ˈpɒlɪɡɒn/) is aplanefiguremade up ofline segmentsconnected to form aclosed polygonal chain.

The segments of a closed polygonal chain are called itsedgesorsides.The points where two edges meet are the polygon'sverticesorcorners.Ann-gonis a polygon withnsides; for example, atriangleis a 3-gon.

Asimple polygonis 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 asolid polygon.The interior of a solid polygon is itsbody,also known as apolygonal regionorpolygonal area.In contexts where one is concerned only with simple and solid polygons, apolygonmay refer only to a simple polygon or to a solid polygon.

A polygonal chain may cross over itself, creatingstar polygonsand otherself-intersecting polygons.Some sources also consider closed polygonal chains inEuclidean spaceto be a type of polygon (askew polygon), even when the chain does not lie in a single plane.

A polygon is a 2-dimensional example of the more generalpolytopein any number of dimensions. There are many moregeneralizations of polygonsdefined for different purposes.

Etymology

The wordpolygonderives from theGreekadjective πολύς (polús) 'much', 'many' and γωνία (gōnía) 'corner' or 'angle'. It has been suggested that γόνυ (gónu) 'knee' may be the origin ofgon.

Classification

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: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.
• 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:the boundary of the polygon does not cross itself. All convex polygons are simple.
• Concave:Non-convex and simple. There is at least one interior angle greater than 180°.
• 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.
• Self-intersecting:the boundary of the polygon crosses itself. The termcomplexis sometimes used in contrast tosimple,but this usage risks confusion with the idea of acomplex polygonas one which exists in the complexHilbertplane consisting of twocomplexdimensions.
• 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:all corner angles are equal.
• Equilateral:all edges are of the same length.
• Regular:both equilateral and equiangular.
• Cyclic:all corners lie on a singlecircle,called thecircumcircle.
• Tangential:all sides are tangent to aninscribed circle.
• Isogonal orvertex-transitive:all corners lie within the samesymmetry orbit.The polygon is also cyclic and equiangular.
• Isotoxal oredge-transitive:all sides lie within the samesymmetry 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 aregularstar polygon.

Miscellaneous

• Rectilinear:the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
• Monotonewith respect to a given lineL:every lineorthogonalto L intersects the polygon not more than twice.

Properties and formulas

Partitioning ann-gon inton− 2triangles

Euclidean geometryis 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 simplen-gon is(n− 2) ×πradiansor(n− 2) × 180degrees.This is because any simplen-gon ( havingnsides ) can be considered to be made up of(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 regularn-gon is$\left(1-{\tfrac {2}{n}}\right)\pi$radians or$180-{\tfrac {360}{n}}$degrees. The interior angles of regularstar polygonswere first studied by Poinsot, in the same paper in which he describes the fourregular star polyhedra:for a regular${\tfrac {p}{q}}$-gon (ap-gon with central densityq), each interior angle is${\tfrac {\pi (p-2q)}{p}}$radians or${\tfrac {180(p-2q)}{p}}$degrees.
• Exterior angle– The exterior angle is thesupplementary angleto the interior angle. Tracing around a convexn-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one fullturn,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 ann-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multipledof 360°, e.g. 720° for apentagramand 0° for an angular "eight" orantiparallelogram,wheredis thedensityorturning numberof the polygon.

Area

Coordinates of a non-convex pentagon

In this section, the vertices of the polygon under consideration are taken to be$(x_{0},y_{0}),(x_{1},y_{1}),\ldots,(x_{n-1},y_{n-1})$in order. For convenience in some formulas, the notation(x_n,y_n) = (x₀,y₀)will also be used.

Simple polygons

Further information:Shoelace formula

If the polygon is non-self-intersecting (that is,simple), the signedareais

$A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0},$

or, usingdeterminants

$16A^{2}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}{\begin{vmatrix}Q_{i,j}&Q_{i,j+1}\\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}},$

where$Q_{i,j}$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 theorientationof the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positivex-axis to the positivey-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 inabsolute value.This is commonly called theshoelace formulaorsurveyor's formula.

The areaAof a simple polygon can also be computed if the lengths of the sides,a₁,a₂,...,a_nand theexterior angles,θ₁,θ₂,...,θ_nare known, from:

${\begin{aligned}A={\frac {1}{2}}(a_{1}[a_{2}\sin(\theta _{1})+a_{3}\sin(\theta _{1}+\theta _{2})+\cdots +a_{n-1}\sin(\theta _{1}+\theta _{2}+\cdots +\theta _{n-2})]\\{}+a_{2}[a_{3}\sin(\theta _{2})+a_{4}\sin(\theta _{2}+\theta _{3})+\cdots +a_{n-1}\sin(\theta _{2}+\cdots +\theta _{n-2})]\\{}+\cdots +a_{n-2}[a_{n-1}\sin(\theta _{n-2})]).\end{aligned}}$

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,Pick's theoremgives 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 perimeterpand areaA,theisoperimetric inequality$p^{2}>4\pi A$holds.

For any two simple polygons of equal area, theBolyai–Gerwien theoremasserts 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 simple and cyclic then the sidesdodetermine the area.Of alln-gons with given side lengths, the one with the largest area is cyclic. Of alln-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 polygons.

The area of a regular polygon is given in terms of the radiusrof itsinscribed circleand its perimeterpby

$A={\tfrac {1}{2}}\cdot p\cdot r.$

This radius is also termed itsapothemand is often represented asa.

The area of a regularn-gon in terms of the radiusRof itscircumscribed circlecan be expressed trigonometrically as:

$A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}}$

The area of a regularn-gon inscribed in a unit-radius circle, with sidesand interior angle$\alpha,$can also be expressed trigonometrically as:

$A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}={\frac {ns^{2}}{4}}\cot {\frac {\alpha }{n-2}}=n\cdot \sin {\frac {\alpha }{n-2}}\cdot \cos {\frac {\alpha }{n-2}}.$

Self-intersecting

The area of aself-intersecting polygoncan 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 thedensityof 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

$C_{x}={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}),$

$C_{y}={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}).$

In these formulas, the signed value of area$A$must be used.

Fortriangles(n= 3), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true forn> 3.Thecentroidof the vertex set of a polygon withnvertices has the coordinates

$c_{x}={\frac {1}{n}}\sum _{i=0}^{n-1}x_{i},$

$c_{y}={\frac {1}{n}}\sum _{i=0}^{n-1}y_{i}.$

Generalizations

The idea of a polygon has been generalized in various ways. Some of the more important include:

• Aspherical polygonis a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows thedigon,a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role incartography(map making) and inWythoff's constructionof theuniform polyhedra.
• Askew polygondoes not lie in a flat plane, but zigzags in three (or more) dimensions. ThePetrie polygonsof the regular polytopes are well known examples.
• Anapeirogonis an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions.
• Askew apeirogonis an infinite sequence of sides and angles that do not lie in a flat plane.
• Apolygon with holesis an area-connected or multiply-connected planar polygon with one external boundary and one or more interior boundaries (holes).
• Acomplex polygonis aconfigurationanalogous to an ordinary polygon, which exists in thecomplex planeof tworealand twoimaginarydimensions.
• Anabstract polygonis an algebraicpartially ordered setrepresenting the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be arealizationof the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized.
• Apolyhedronis 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 calledpolytopes.(In other conventions, the wordspolyhedronandpolytopeare used in any dimension, with the distinction between the two that a polytope is necessarily bounded.)

Naming

The wordpolygoncomes fromLate Latinpolygōnum(a noun), fromGreekπολύγωνον (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 aGreek-derivednumerical prefixwith the suffix-gon,e.g.pentagon,dodecagon.Thetriangle,quadrilateralandnonagonare exceptions.

Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.

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 theregularstarpentagonis also known as thepentagram.

Polygon names and miscellaneous properties

Name	Sides	Properties
monogon	1	Not generally recognised as a polygon,although some disciplines such as graph theory sometimes use the term.
digon	2	Not generally recognised as a polygon in the Euclidean plane, although it can exist as aspherical polygon.
triangle(or trigon)	3	The simplest polygon which can exist in the Euclidean plane. Cantilethe 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. Cantilethe plane.
pentagon	5	The simplest polygon which can exist as a regular star. A star pentagon is known as apentagramor pentacle.
hexagon	6	Cantilethe plane.
heptagon(or septagon)	7	The simplest polygon such that the regular form is notconstructiblewithcompass and straightedge.However, it can be constructed using aneusis construction.
octagon	8
nonagon(or enneagon)	9	"Nonagon" mixes Latin [novem= 9] with Greek; "enneagon" is pure Greek.
decagon	10
hendecagon(or undecagon)	11	The simplest polygon such that the regular form cannot be constructed with compass, straightedge, andangle trisector.However, it can be constructed with neusis.
dodecagon(or duodecagon)	12
tridecagon(or triskaidecagon)	13
tetradecagon(or tetrakaidecagon)	14
pentadecagon(or pentakaidecagon)	15
hexadecagon(or hexakaidecagon)	16
heptadecagon(or heptakaidecagon)	17	Constructible polygon
octadecagon(or octakaidecagon)	18
enneadecagon (or enneakaidecagon)	19
icosagon	20
icositrigon(or icosikaitrigon)	23	The simplest polygon such that the regular form cannot be constructed withneusis.
icositetragon(or icosikaitetragon)	24
icosipentagon (or icosikaipentagon)	25	The simplest polygon such that it is not known if the regular form can be constructed with neusis or not.
triacontagon	30
tetracontagon (or tessaracontagon)	40
pentacontagon (or pentecontagon)	50
hexacontagon (or hexecontagon)	60
heptacontagon (or hebdomecontagon)	70
octacontagon (or ogdoëcontagon)	80
enneacontagon (or enenecontagon)	90
hectogon (or hecatontagon)	100
257-gon	257	Constructible polygon
chiliagon	1000	Philosophers includingRené Descartes,Immanuel Kant,David Hume,have used the chiliagon as an example in discussions.
myriagon	10,000	Used as an example in some philosophical discussions, for example in Descartes'sMeditations on First Philosophy
65537-gon	65,537	Constructible polygon
megagon	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.The megagon is also used as an illustration of the convergence ofregular polygonsto a circle.
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.The "kai" term applies to 13-gons and higher and was used byKepler,and advocated byJohn H. Conwayfor clarity of concatenated prefix numbers in the naming ofquasiregular polyhedra,though not all sources use it.

Tens		and	Ones		final suffix
		-kai-	1	-hena-	-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

Historical image of polygons (1699)

Polygons have been known since ancient times. Theregular polygonswere known to the ancient Greeks, with thepentagram,a non-convex regular polygon (star polygon), appearing as early as the 7th century B.C. on akraterbyAristophanes,found atCaereand now in theCapitoline Museum.

The first known systematic study of non-convex polygons in general was made byThomas Bradwardinein the 14th century.

In 1952,Geoffrey Colin Shephardgeneralized the idea of polygons to the complex plane, where eachrealdimension is accompanied by animaginaryone, to createcomplex polygons.

In nature

TheGiant's Causeway,inNorthern Ireland

Polygons appear in rock formations, most commonly as the flat facets ofcrystals,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 oflavaforms areas of tightly packed columns ofbasalt,which may be seen at theGiant's CausewayinNorthern Ireland,or at theDevil's PostpileinCalifornia.

Inbiology,the surface of the waxhoneycombmade bybeesis an array ofhexagons,and the sides and base of each cell are also polygons.

Computer graphics

Main article:Polygon (computer graphics)

Incomputer graphics,a polygon is aprimitiveused in modelling and rendering. They are defined in a database, containing arrays ofvertices(the coordinates of thegeometrical vertices,as well as other attributes of the polygon, such as color, shading and texture), connectivity information, andmaterials.

Any surface is modelled as a tessellation calledpolygon mesh.If a square mesh hasn+ 1points (vertices) per side, there arensquared squares in the mesh, or 2nsquared triangles since there are two triangles in a square. There are(n+ 1)²/ 2(n²)vertices per triangle. Wherenis 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 andcomputational 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. This is called thepoint in polygontest.

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
• Tiling
• Tiling puzzle

References

Bibliography

• Coxeter, H.S.M.;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. (pdf)

External links

• Weisstein, Eric W."Polygon".MathWorld.
• What Are Polyhedra?,with Greek Numerical Prefixes
• Polygons, types of polygons, and polygon properties,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
• Comparison of the different algorithms for Polygon Boolean operations,compares capabilities, speed and numerical robustness
• 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