In geometry, a polygon ( // ) 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 bounded 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. 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 is a 3-gon.
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 self-intersecting polygons.
A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes.
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.
Polygons are primarily classified by the number of sides. See the table below.
Polygons may be characterized by their convexity or type of non-convexity:
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 .
Euclidean geometry is assumed throughout.
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:
In this section, the vertices of the polygon under consideration are taken to be in order. For convenience in some formulas, the notation (xn, yn) = (x0, y0) will also be used.
If the polygon is non-self-intersecting (that is, simple), the signed area is
or, using determinants
where is the squared distance between and
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 x-axis to the positive 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.
The area A of a simple polygon can also be computed if the lengths of the sides, a1, a2, ..., an and the exterior angles, θ1, θ2, ..., θn are known, from:
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 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 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 simple and 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).
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 n-gon in terms of the radius R of its circumscribed circle can be expressed trigonometrically as:
The area of a regular n-gon inscribed in a unit-radius circle, with side s and interior angle can also be expressed trigonometrically as:
The area of a self-intersecting polygon can be defined in two different ways, giving different answers:
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
The idea of a polygon has been generalized in various ways. Some of the more important include:
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 "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a 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.
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 star pentagon is also known as the pentagram.
|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 a spherical polygon.|
|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 non-cyclic. Can tile the plane.|
|pentagon||5||The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle.|
|hexagon||6||Can tile the plane.|
|heptagon (or septagon)||7||The simplest polygon such that the regular form is not constructible with compass and straightedge. However, it can be constructed using a neusis construction.|
|nonagon (or enneagon)||9||"Nonagon" mixes Latin [novem = 9] with Greek; "enneagon" is pure Greek.|
|hendecagon (or undecagon)||11||The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and angle 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|
|icositrigon (or icosikaitrigon)||23||The simplest polygon such that the regular form cannot be constructed with neusis.|
|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.|
|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|
|chiliagon||1000||Philosophers including René 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's Meditations on First Philosophy|
|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 of regular polygons to a circle.|
|apeirogon||∞||A degenerate polygon of infinitely many sides.|
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 Kepler, and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra, though not all sources use it.
|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-|
Polygons have been known since ancient times. The regular polygons 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, 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 real dimension is accompanied by an imaginary one, to create complex polygons.
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.
This section needs additional citations for verification .(October 2018)
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.
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(n2) 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 lies inside a simple polygon given by a sequence of line segments. This is called the point in polygon test.
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope made with line segments in any given dimension.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus.
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is also known as the arc's angular distance.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.
In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
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.
In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.
Two-dimensional Euclidean space or simply two-dimensional space is a geometric setting in which two values are required to determine the position of an element on the plane. It is the two-dimensional version of Euclidean space.
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
|Family||An||Bn||I2(p) / Dn||E6 / E7 / E8 / F4 / G2||Hn|
|Uniform polyhedron||Tetrahedron||Octahedron • Cube||Demicube||Dodecahedron • Icosahedron|
|Uniform polychoron||Pentachoron||16-cell • Tesseract||Demitesseract||24-cell||120-cell • 600-cell|
|Uniform 5-polytope||5-simplex||5-orthoplex • 5-cube||5-demicube|
|Uniform 6-polytope||6-simplex||6-orthoplex • 6-cube||6-demicube||122 • 221|
|Uniform 7-polytope||7-simplex||7-orthoplex • 7-cube||7-demicube||132 • 231 • 321|
|Uniform 8-polytope||8-simplex||8-orthoplex • 8-cube||8-demicube||142 • 241 • 421|
|Uniform 9-polytope||9-simplex||9-orthoplex • 9-cube||9-demicube|
|Uniform 10-polytope||10-simplex||10-orthoplex • 10-cube||10-demicube|
|Uniform n-polytope||n-simplex||n-orthoplex • n-cube||n-demicube||1k2 • 2k1 • k21||n-pentagonal polytope|
|Topics: Polytope families • Regular polytope • List of regular polytopes and compounds|