An assortment of polygons.

In geometry, a polygon is a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (that is, by a closed polygonal chain). These line segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. The interior of the polygon is called its body. A polygon is a 2-dimensional example of the more general "polytope" in any number of dimensions.




Number of sides



Polygons may be characterized by their degree of convexity:

  • Concave: Non-convex and simple.
  • Self-intersecting: the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this is mistaken: a complex polygon is one which exists in the unitary plane, which comprises two complex dimensions.


  • Equiangular: all its corner angles are equal.
  • Cyclic: all corners lie on a single circle.
  • Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
  • Equilateral: all edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex.) (Williams, 1979, 31-32)
  • Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral.
  • Regular. A polygon is regular if it is both cyclic and equilateral.'. A non-convex regular polygon is called a regular star polygon.




We will assume Euclidean geometry throughout.


  • Any polygon, regular or irregular, complex or simple, 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 (n−2)π radians or (n−2)180 degrees. This is because 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 degrees. In topology and analysis,
    • Exterior angle - Imagine walking around a simple n-gon marked on the floor. The amount you "turn" at a corner is the exterior or external angle. Walking all the way round the polygon, you make one full turn, so the sum of the exterior angles must be 360°. The exterior angle is the supplementary angle to the interior angle, and from this the sum of the interior angles can be easily confirmed.


The measure of any interior angle of a convex regular n-gon is (n−2)π/n radians or (n−2)180/n 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.

Moving around an n-gon in general, the sum of the exterior angles (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).

Area and centroid

Nomenclature of a 2D polygon.



To close the polygon, the first and last vertices are the same, ie x_n, y_n = x_0, y_0. The vertices must be ordered clockwise or counterclockwise, if they are ordered clockwise the area will be negative but correct in absolute value.

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.

The area A of a simple polygon can also be computed if the lengths of the sides, a1,a2, …, an and the exterior angles, \theta_1, \theta_2, …, \theta_n are known. The formula is


The formula was described by Lopshits in 1963.[2]





Self-intersecting polygons



Degrees of freedom

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


Generalizations of polygons

In a broad sense, a polygon is an unbounded sequence or circuit of alternating segments (sides) and angles (corners). The modern mathematical understanding is to describe this structural sequence in terms of an 'abstract' polygon which is a partially-ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.

Generally, a geometric polygon is a 'realization' of this abstract polygon; this involves some 'mapping' of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. As another example, most polygons are unbounded because they close back on themselves, while apeirogons (infinite polygons) are unbounded because they go on for ever so you can never reach any bounding end point. So when we talk about "polygons" we must be careful to explain what kind we are talking about.




  • Digon. Angle of 0° in the Euclidean plane. See remarks above re. on the sphere.
  • Angle of 180°: In the plane this gives an apeirogon), on the sphere a dihedron
  • A complex polygon is a figure analogous to an ordinary polygon, which exists in the unitary plane.

Naming polygons

The word 'polygon' comes from Late Latin polygōnum (a noun), from Greek polygōnon/polugōnon πολύγωνον, noun use of neuter of polygōnos/polugōnos πολύγωνος (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. For large numbers, mathematicians usually 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.

Some special polygons also have their own names; for example, the regular star pentagon is also known as the pentagram.

Polygon names
Name Edges
henagon (or monogon) 1
digon 2
triangle (or trigon) 3
quadrilateral (or tetragon) 4
pentagon 5
hexagon 6
heptagon (avoid "septagon" = Latin [sept-] + Greek) 7
octagon 8
enneagon (or nonagon) 9
decagon 10
hendecagon (avoid "undecagon" = Latin [un-] + Greek) 11
dodecagon (avoid "duodecagon" = Latin [duo-] + Greek) 12
tridecagon (or triskaidecagon) 13
tetradecagon (or tetrakaidecagon) 14
pentadecagon (or quindecagon or pentakaidecagon) 15
hexadecagon (or hexakaidecagon) 16
heptadecagon (or heptakaidecagon) 17
octadecagon (or octakaidecagon) 18
enneadecagon (or enneakaidecagon or nonadecagon) 19
icosagon 20
No established English name

"hectogon" is the Greek name (see hectometre),

chiliagon 1000
myriagon 10,000
googolgon 10100


Tens and Ones final suffix
-kai- 1 -hena- -gon
20 icosi- 2 -di-
30 triaconta- 3 -tri-
40 tetraconta- 4 -tetra-
50 pentaconta- 5 -penta-
60 hexaconta- 6 -hexa-
70 heptaconta- 7 -hepta-
80 octaconta- 8 -octa-
90 enneaconta- 9 -ennea-



Tens and Ones final suffix full polygon name
tetraconta- -kai- -di- -gon tetracontakaidigon

and a 50-sided figure

Tens and Ones final suffix full polygon name
pentaconta-   -gon pentacontagon

But beyond enneagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons).

Polygons in nature

The Giant's Causeway, in Ireland

Numerous regular polygons may be seen in nature. In the world of minerals, crystals often have faces which are triangular, square or hexagonal. Quasicrystals can even have regular pentagons as faces. Another fascinating example of regular polygons occurs when the cooling of lava forms areas of tightly packed hexagonal columns of basalt, which may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California.

Starfruit, a popular fruit in Southeast Asia




Things to do with polygons


Polygons in computer graphics


Naming conventions differ from those of mathematicians:

  • A simple polygon does not cross itself.
  • A concave polygon is a simple polygon having at least one interior angle greater than 180 deg.
  • A complex polygon does cross itself.



Polygon Count. Since a polygon can have many sides and need many points to define it, in order to compare one imaging system with another, "polygon count" is generally taken as a triangle. A triangle is processed as three points in the x, y, and z axes, needing nine geometrical descriptors. In addition, coding is applied to each polygon for color, brightness, shading, texture, NVG (intensifier or night vision), Infra-Red characteristics and so on. When analyzing the characteristics of a particular imaging system, the exact definition of polygon count should be obtained as it applies to that system.




See also


  1. Paul Bourke. 1988. Polygon Area and Centroid. University of Western Australia. Retrieved November 3, 2007.
  2. A.M. Lopshits, J. Massalski and C. Mills, Jr. trans. 1963. Computation of areas of oriented figures. (Boston, MA: DC Heath and Company, 1963.)


  • Stillwell, John. 2005. The Four Pillars of Geometry. Undergraduate Texts in Mathematics. New York: Springer. ISBN 0387255303.

External links

All links retrieved March 31, 2019.




