Polygon
In elementary geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain 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 ngon is a polygon with n sides; for example, a triangle is a 3gon.
A simple polygon 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 polygons and other selfintersecting polygons.
A polygon is a 2dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes.
Contents
Etymology
The word polygon derives from the 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.^{[1]}
Classification
Number of sides
Polygons are primarily classified by the number of sides. See the table below.
Convexity and nonconvexity
Polygons may be characterized by their convexity or type of nonconvexity:
 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.
 Nonconvex: 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. Nonconvex and simple. There is at least one interior angle greater than 180°.
 Starshaped: 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 starshaped.
 Selfintersecting: 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 as one which exists in the complex Hilbert plane consisting of two complex dimensions.
 Star polygon: a polygon which selfintersects in a regular way. A polygon cannot be both a star and starshaped.
Equality and symmetry
 Equiangular: all corner angles are equal.
 Cyclic: all corners lie on a single circle, called the circumcircle.
 Isogonal or vertextransitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
 Equilateral: all edges are of the same length. The polygon need not be convex.
 Tangential: all sides are tangent to an inscribed circle.
 Isotoxal or edgetransitive: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential.
 Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both cyclic and equilateral, or both equilateral and equiangular. A nonconvex regular polygon is called a regular star polygon.
Miscellaneous
 Rectilinear: 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 to L intersects the polygon not more than twice.
Properties and formulas
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 ngon is (n − 2)π radians or (n − 2) × 180 degrees. This is because any simple ngon ( having n sides ) 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 regular ngon is radians or degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular gon (a pgon with central density q), each interior angle is radians or degrees.^{[2]}
 Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex ngon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full 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 ngon 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 or starriness of the polygon. See also orbit (dynamics).
Area
In this section, the vertices of the polygon under consideration are taken to be in order. For convenience in some formulas, the notation (x_{n}, y_{n}) = (x_{0}, y_{0}) will also be used.
If the polygon is nonselfintersecting (that is, simple), the signed area is
or, using determinants
where is the squared distance between and ^{[3]}^{[4]}
The signed area depends on the ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive xaxis to the positive yaxis. 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.^{[5]}
The area A of a simple polygon can also be computed if the lengths of the sides, a_{1}, a_{2}, ..., a_{n} and the exterior angles, θ_{1}, θ_{2}, ..., θ_{n} are known, from:
The formula was described by Lopshits in 1963.^{[6]}
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 onehalf the latter number, minus 1.
In every polygon with perimeter p and area A , the isoperimetric inequality holds.^{[7]}
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.^{[8]} However, if the polygon is cyclic then the sides do determine the area.^{[citation needed]} Of all ngons with given side lengths, the one with the largest area is cyclic. Of all ngons with a given perimeter, the one with the largest area is regular (and therefore cyclic).^{[9]}
Regular polygons
Many specialized formulas apply to the areas of regular polygons.
The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by
This radius is also termed its apothem and is often represented as a.
The area of a regular ngon with side s inscribed in a unit circle is
The area of a regular ngon in terms of the radius R of its circumscribed circle and its perimeter p is given by
The area of a regular ngon inscribed in a unitradius circle, with side s and interior angle can also be expressed trigonometrically as
Selfintersecting
The area of a selfintersecting 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 crossquadrilateral (like a figure 8) have oppositesigned densities, and adding their areas together can give a total area of zero for the whole figure.^{[citation needed]}
 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 selfintersecting one. In the case of the crossquadrilateral, it is treated as two simple triangles.^{[citation needed]}
Centroid
Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are
In these formulas, the signed value of area must be used.
For triangles (n = 3), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3. The centroid of the vertex set of a polygon with n vertices has the coordinates
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 polygons 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 complex polygon is a configuration analogous to an ordinary polygon, which exists in the complex plane of two real and two imaginary dimensions.
 An 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 threedimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are called polytopes.^{[10]} (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.^{[11]})
Naming
The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "manyangled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greekderived numerical prefix with the suffix gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions.
Beyond decagons (10sided) and dodecagons (12sided), mathematicians generally use numerical notation, for example 17gon and 257gon.^{[12]}
Exceptions exist for side counts that are more easily expressed in verbal form (e.g. 20 and 30), or are used by nonmathematicians. Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram.
Name  Edges  Properties 

monogon  1  Not generally recognised as a polygon,^{[13]} although some disciplines such as graph theory sometimes use the term.^{[14]} 
digon  2  Not generally recognised as a polygon in the Euclidean plane, although it can exist as a spherical polygon.^{[15]} 
triangle (or trigon)  3  The simplest polygon which can exist in the Euclidean plane. Can 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 noncyclic. Can tile the plane. 
pentagon  5  ^{[16]} The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle. 
hexagon  6  ^{[16]} Can tile the plane. 
heptagon (or septagon)  7  ^{[16]} The simplest polygon such that the regular form is not constructible with compass and straightedge. However, it can be constructed using a Neusis construction. 
octagon  8  ^{[16]} 
nonagon (or enneagon)  9  ^{[16]}"Nonagon" mixes Latin [novem = 9] with Greek, "enneagon" is pure Greek. 
decagon  10  ^{[16]} 
hendecagon (or undecagon)  11  ^{[16]} The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and angle trisector. 
dodecagon (or duodecagon)  12  ^{[16]} 
tridecagon (or triskaidecagon)  13  ^{[16]} 
tetradecagon (or tetrakaidecagon)  14  ^{[16]} 
pentadecagon (or pentakaidecagon)  15  ^{[16]} 
hexadecagon (or hexakaidecagon)  16  ^{[16]} 
heptadecagon (or heptakaidecagon)  17  Constructible polygon^{[12]} 
octadecagon (or octakaidecagon)  18  ^{[16]} 
enneadecagon (or enneakaidecagon)  19  ^{[16]} 
icosagon  20  ^{[16]} 
icositetragon (or icosikaitetragon)  24  ^{[16]} 
triacontagon  30  ^{[16]} 
tetracontagon (or tessaracontagon)  40  ^{[16]}^{[17]} 
pentacontagon (or pentecontagon)  50  ^{[16]}^{[17]} 
hexacontagon (or hexecontagon)  60  ^{[16]}^{[17]} 
heptacontagon (or hebdomecontagon)  70  ^{[16]}^{[17]} 
octacontagon (or ogdoëcontagon)  80  ^{[16]}^{[17]} 
enneacontagon (or enenecontagon)  90  ^{[16]}^{[17]} 
hectogon (or hecatontagon)^{[18]}  100  ^{[16]} 
257gon  257  Constructible polygon^{[12]} 
chiliagon  1000  Philosophers including René Descartes,^{[19]} Immanuel Kant,^{[20]} David Hume,^{[21]} have used the chiliagon as an example in discussions. 
myriagon  10,000  Used as an example in some philosophical discussions, for example in Descartes' Meditations on First Philosophy 
65537gon  65,537  Constructible polygon^{[12]} 
megagon^{[22]}^{[23]}^{[24]}  1,000,000  As with René Descartes' example of the chiliagon, the millionsided polygon has been used as an illustration of a welldefined concept that cannot be visualised.^{[25]}^{[26]}^{[27]}^{[28]}^{[29]}^{[30]}^{[31]} The megagon is also used as an illustration of the convergence of regular polygons to a circle.^{[32]} 
apeirogon  ∞  A degenerate polygon of infinitely many sides. 
Constructing higher names
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows.^{[16]} The "kai" term applies to 13gons and higher and was used by Kepler, and advocated by John H. Conway for clarity to concatenated prefix numbers in the naming of quasiregular polyhedra.^{[18]}
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
Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with the pentagram, a nonconvex regular polygon (star polygon), appearing as early as the 7th century B.C. on a krater by Aristophanes, found at Caere and now in the Capitoline Museum.^{[33]}^{[34]}
The first known systematic study of nonconvex polygons in general was made by Thomas Bradwardine in the 14th century.^{[35]}
In 1952, Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.^{[36]}
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
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In computer graphics, a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of the geometrical vertices, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials.^{[37]}^{[38]}
Naming conventions differ from those of mathematicians:^{[clarification needed (What is the difference?)]}
 A simple polygon is convex, planar, more easily handled by algorithms and hardware.
 a concave polygon is a simple polygon having at least one interior angle greater than 180°.
 A complex polygon may have arbitrary topology including holes, requiring more advanced algorithms (often systems will tesselate these into simple polygons).^{[citation needed]}
Any surface is modelled as a tessellation called polygon mesh. If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. There are (n + 1)^{2} / 2(n^{2}) 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 twodimensional, through the system computer they are placed in a visual scene in the correct threedimensional orientation.
In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x_{0},y_{0}) lies inside a simple polygon given by a sequence of line segments. This is called the Point in polygon test.^{[citation needed]}
See also
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 GoodmanPollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)
Notes
 ^ Craig, John (1849). A new universal etymological technological, and pronouncing dictionary of the English language. Oxford University. p. 404. Extract of p. 404
 ^ Kappraff, Jay (2002). Beyond measure: a guided tour through nature, myth, and number. World Scientific. p. 258. ISBN 9789810247027.
 ^ B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949)
 ^ Bourke, Paul (July 1988). "Calculating The Area And Centroid Of A Polygon" (PDF). Retrieved 6 Feb 2013.
 ^ Bart Braden (1986). "The Surveyor's Area Formula" (PDF). The College Mathematics Journal. 17 (4): 326–337. doi:10.2307/2686282.
 ^ A.M. Lopshits (1963). Computation of areas of oriented figures. translators: J Massalski and C Mills, Jr. D C Heath and Company: Boston, MA.
 ^ Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality", Forum Mathematicorum 2, 2002, 129–130.
 ^ Robbins, "Polygons inscribed in a circle," American Mathematical Monthly 102, June–July 1995.
 ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
 ^ Coxeter (3rd Ed 1973)
 ^ Günter Ziegler (1995). "Lectures on Polytopes". Springer Graduate Texts in Mathematics, ISBN 9780387943657. p. 4.
 ^ ^{a} ^{b} ^{c} ^{d} Mathworld
 ^ Grunbaum, B.; "Are your polyhedra the same as my polyhedra", Discrete and computational geometry: the GoodmanPollack Festschrift, Ed. Aronov et al., Springer (2003), p. 464.
 ^ Hass, Joel; Morgan, Frank (1996), "Geodesic nets on the 2sphere", Proceedings of the American Mathematical Society, 124 (12): 3843–3850, doi:10.1090/S0002993996034922, JSTOR 2161556, MR 1343696.
 ^ Coxeter, H.S.M.; Regular polytopes, Dover Edition (1973), p. 4.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} ^{r} ^{s} ^{t} ^{u} ^{v} ^{w} ^{x} ^{y} Salomon, David (2011). The Computer Graphics Manual. Springer Science & Business Media. pp. 88–90. ISBN 9780857298867.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
 ^ ^{a} ^{b} "Naming Polygons and Polyhedra". Ask Dr. Math. The Math Forum – Drexel University. Retrieved 3 May 2015.
 ^ Sepkoski, David (2005). "Nominalism and constructivism in seventeenthcentury mathematical philosophy" (PDF). Historia Mathematica. 32: 33–59. doi:10.1016/j.hm.2003.09.002. Archived from the original (PDF) on 12 May 2012. Retrieved 18 April 2012.
 ^ Gottfried Martin (1955), Kant's Metaphysics and Theory of Science, Manchester University Press, p. 22.
 ^ David Hume, The Philosophical Works of David Hume, Volume 1, Black and Tait, 1826, p. 101.
 ^ Gibilisco, Stan (2003). Geometry demystified (OnlineAusg. ed.). New York: McGrawHill. ISBN 9780071416504.
 ^ Darling, David J., The universal book of mathematics: from Abracadabra to Zeno's paradoxes, John Wiley & Sons, 2004. p. 249. ISBN 0471270474.
 ^ Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, AddisonWesley, 1999. p. 505. ISBN 0201347121.
 ^ McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
 ^ Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0582281571.
 ^ Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0415157927.
 ^ Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1847063497.
 ^ Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0198752776.
 ^ Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
 ^ Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0823214869.
 ^ Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0415325056.
 ^ Heath, Sir Thomas Little (1981), A History of Greek Mathematics, Volume 1, Courier Dover Publications, p. 162, ISBN 9780486240732. Reprint of original 1921 publication with corrected errata. Heath uses the Latinized spelling "Aristophonus" for the vase painter's name.
 ^ Cratere with the blinding of Polyphemus and a naval battle Archived 20131112 at the Wayback Machine, Castellani Halls, Capitoline Museum, accessed 20131111. Two pentagrams are visible near the center of the image,
 ^ Coxeter, H.S.M.; Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p. 114
 ^ Shephard, G.C.; "Regular complex polytopes", Proc. London Math. Soc. Series 3 Volume 2, 1952, pp 8297
 ^ "opengl vertex specification".
 ^ "direct3d rendering, based on vertices & triangles".
External links
Look up polygon in Wiktionary, the free dictionary. 
Wikimedia Commons has media related to Polygons. 
 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