Look up Polygon on Wiktionary, the free dictionary

For other use please see Polygon (disambiguation)

A polygon (literally "many angle", see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices. If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary. Wikipedia does not have an article with this exact name. ... Jump to: navigation, search Logo en:Wiktionary Wiktionary is a sister project to Wikipedia intended to be a free wiki dictionary (including thesaurus and lexicon) in every language. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... For other uses, see Curve (disambiguation). ... In mathematics, a plane is the fundamental two-dimensional object. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... The word Boundary has a variety of meanings. ...

Names and types

A simple concave hexagon

A complex pentagon

Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula. A simple polygon, created with GIMP, in the public domain. ... A simple polygon, created with GIMP, in the public domain. ... A complex polygon, created with GIMP, in the public domain. ... A complex polygon, created with GIMP, in the public domain. ... A numerical prefix is a prefix that denotes a number, which is usually a multiplier for the thing being prefixed. ... Jump to: navigation, search In geometry, a pentagon is any five-sided polygon. ... Jump to: navigation, search A regular dodecagon A dodecagon is a polygon with exactly twelve sides. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Other uses: Quadrilateral (disambiguation) In geometry, a quadrilateral is a polygon with four sides and four vertices. ... A mathematician is a person whose area of study and research is mathematics. ... A numeral is a symbol or group of symbols that represents a number. ...

For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Other uses: Quadrilateral (disambiguation) In geometry, a quadrilateral is a polygon with four sides and four vertices. ... Jump to: navigation, search In geometry, a pentagon is any five-sided polygon. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... A heptagon is a plane figure with seven sides and seven angles. ... Jump to: navigation, search One of the 8 semi-regular tessellations: octagons and squares An octagon is a polygon that has eight sides. ... In geometry, an enneagon or nonagon is a nine-sided polygon. ... An image of a Regular Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144°. Its Schläfli symbol is {10}. The area of a regular decagon... Categories: Math stubs | Polygons ... Jump to: navigation, search A regular dodecagon A dodecagon is a polygon with exactly twelve sides. ... A triskaidecagon is a polygon with 13 sides and angles. ... In geometry, a pentadecagon is any 15-sided, 15-angled, polygon. ... In geometry, a heptadecagon is a seventeen-sided polygon. ... Categories: Math stubs | Polygons ... In geometry, an icosagon is a twenty-sided polygon. ... A tricontagon is a polygon with 30 sides. ... A pentacontagon is a polygon with 50 sides. ... A hectagon is a polygon with 100 edges. ... A chiliagon is a polygon with 1000 sides. ... A myriagon is a polygon with 10,000 sides. ...

Naming polygons

To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows

That is, a 42-sided figure would be named as follows:

and a 50-sided figure

But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Taxonomic classification

The taxonomic classification of polygons is illustrated by the following tree:

 Polygon /  Simple Complex /  Convex Concave / Cyclic / Regular 

• A polygon is called simple if it is described by a single, non-intersecting boundary (hence has an inside and an outside); otherwise it is called complex.
• A simple polygon is called convex if it has no internal angles greater than 180°; otherwise it is called concave.
• A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
• A cyclic polygon is called regular if all its sides are of equal length and all its angles are equal; for each number of sides, all regular polygons are similar.

The regular polygons include: A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... Categories: Stub | Polygons ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... Jump to: navigation, search In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... Several equivalence relations in mathematics are called similarity. ...

• equilateral triangle
• square
• regular pentagon
• regular hexagon

See also tilings of regular polygons. Jump to: navigation, search A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... Jump to: navigation, search In geometry, a pentagon is any five-sided polygon. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

Properties

We will assume Euclidean geometry throughout. Jump to: navigation, search In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape. The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. ...

In the case of a line of symmetry the latter reduces to n-2.

Let k≥2. For an nk-gon with k-fold rotational symmetry (C_k), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (D_k) there are n-1 degrees of freedom.

Angles

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) turns). This can be seen in two different ways: This article is about angles in geometry. ... Jump to: navigation, search Lower-case pi The mathematical constant Ï€ is the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. ... See Radian (band) for the Austrian trio. ... Turn (geometry) The British Astronomer and Science writer Sir Fred Hoyle (1915 - 2001) in Astronomy (London 1962), proposed dividing the circle, or turn into 1000 milliturns. ...

• Moving around a simple n-gon (like a car on a road), the amount one "turns" at a vertex is 180° minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between -1/2 and 1/2 winding.)
• Any simple n-gon can be considered to be made up of (n−2) triangles, each of which has an angle sum of π radians or 180°.

Moving around an n-gon in general, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a pentagram and 0° for an angular "eight". See also orbit (dynamics). A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ... Jump to: navigation, search A pentagram, pentacle, pentalpha, or pentangle A pentagram is a five-pointed star drawn with five straight strokes. ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ...

Area

Apothem of a hexagon

Several formulae give the area of a regular polygon: Image File history File links Apothem of an hexagon I made this image File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Apothem of an hexagon I made this image File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ...

half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side)

The area A of a simple polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is Area is a quantity expressing the size of a figure in Euclidean plane or surface. ... Jump to: navigation, search Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

A = ½ · (x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + ... + x_ny₁ − x₁y_n)

= ½ · (x₁(y₂ − y_n) + x₂(y₃ − y₁) + x₃(y₄ − y₂) + ... + x_n(y₁ − y_n−1))

The formula was described by Meister in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. Jump to: navigation, search Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special case of the more...

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. Given a simple polygon constructed on a grid of equal-distanced points (i. ...

If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem. In geometry, the Bolyai-Gerwien theorem states that if two simple polygons of equal area are given, one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon. ...

Construction

All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle). In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...

A regular n-sided polygon can be constructed with ruler and compass if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon. A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... In mathematics, any integer (whole number) is either even or odd. ... Jump to: navigation, search In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ...

Point in polygon test

In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x₀,y₀) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test. Jump to: navigation, search Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ... Jump to: navigation, search In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. ... In computational geometry, the point in polygon (also point-in-polygon or PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a simple polygon. ...

Special cases

Some special cases are:

• angle of 0° or 180° (degenerate case)
• two non-adjacent sides are on the same line
• equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32)
• equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32)

A triangle is equilateral iff it is equiangular. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "angular eight" with vertices on a rectangle. Other uses: Quadrilateral (disambiguation) In geometry, a quadrilateral is a polygon with four sides and four vertices. ... Jump to: navigation, search This shape is a rhombus In geometry, a rhombus (also known as a rhomb) is a quadrilateral in which all of the sides are of equal length. ... Jump to: navigation, search In geometry, a rectangle is defined as a quadrilateral polygon in which all four angles are right angles. ...

See also

• cyclic polygon
• geometric shape
• polyform
• polyhedron
• polytope
• simple polygon
• star polygon
• synthetic geometry
• tiling
• tiling puzzle