, a polygon () is a plane figure
that is described by a finite number of straight line segment
s 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 ''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 polygon
s 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''.
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:
: 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.
: the boundary of the polygon does not cross itself. All convex polygons are simple.
: Non-convex and simple. There is at least one interior angle greater than 180°.
: 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.
: 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
* Star polygon
: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.
Equality and symmetry
: 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 ''regular star polygon
* Isogonal or vertex-transitive
: all corners lie within the same symmetry orbit
. The polygon is also cyclic and equiangular.
* Isotoxal or edge-transitive
: all sides lie within the same symmetry orbit
. The polygon is also equilateral and tangential.
: all corners lie on a single circle
, called the circumcircle
: all edges are of the same length. The polygon need not be convex.
: all corner angles are equal.
: all sides are tangent to an inscribed circle
: the polygon's sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.
with respect to a given line ''L'': every line orthogonal
to L intersects the polygon not more than twice.
Properties and formulas
is assumed throughout.
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 ''n''-gon is radian
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
degrees. The interior angles of regular star polygon
s were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra
: for a regular
-gon (a ''p''-gon with central density ''q''), each interior angle is
* Exterior angle
– The exterior angle is the supplementary angle
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
, 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
and 0° for an angular "eight" or antiparallelogram
, where ''d'' is the density
or turning number
of the polygon. See also orbit (dynamics)
In this section, the vertices of the polygon under consideration are taken to be
in order. For convenience in some formulas, the notation will also be used.
If the polygon is non-self-intersecting (that is, simple
), the signed area
or, using determinant
is the squared distance between
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 -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
. 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, ''a''1
, ..., ''an
'' and the exterior angle
, ..., ''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
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).
Many specialized formulas apply to the areas of regular polygon
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 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.
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.
s (), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for . The centroid
of the vertex set of a polygon with vertices has the coordinates
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 polygon
s 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
* 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 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
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.)
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
''. The triangle
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 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 star pentagon
is also known as the pentagram
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 Kepler, and advocated by John H. Conway for clarity to concatenated prefix numbers in the naming of quasiregular polyhedra.]
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.
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 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''0,''y''0) lies inside a simple polygon given by a sequence of line segments. This is called the point in polygon test.
* Boolean operations on polygons
* Complete graph
* Constructible polygon
* Cyclic polygon
* Geometric shape
* List of polygons
* Polygon soup
* Polygon triangulation
* Precision polygon
* Synthetic geometry
* Tiling puzzle
* 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.''
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
Category:Euclidean plane geometry