1 Definitions

We follow Forder [2] for most of this entry. The term polygonMathworldPlanetmathPlanetmath can be defined if one has a definition of an intervalMathworldPlanetmath. For this entry we use betweenness geometryMathworldPlanetmathPlanetmath. A betweenness geometry is just one for which there is a set of points and a betweenness relation B defined. Rather than write (a,b,c)B we write a*b*c.

  1. 1.

    If a and b are distinct points, the line ab is the set of all points p such that p*a*b or a*p*b or a*b*p. It can be shown that the line ab and the line ba are the same set of points.

  2. 2.

    If o and a are distinct points, a ray [oa is the set of all points p such that p=o or o*p*a or o*a*p.

  3. 3.

    If a and b are distinct points, the open interval is the set of points p such that a*p*b. It is denoted by (a,b).

  4. 4.

    If a and b are distinct points, the closed interval is (a,b){a}{b}, and denoted by [a,b].

  5. 5.

    The way a1a2an is the finite setMathworldPlanetmath of points {a1,,an} along with the open intervals (a1,a2),(a2,a3),,(an-1,an). The points a1,,an are called the vertices of the way, and the open intervals are called the sides of the way. A way is also called a broken line. The closed intervals [a1,a2],,[an-1,an] are called the side-intervals of the way. The lines a1a2,,an-1an are called the side-lines of the way. The way a1a2an is said to join a1 to an. It is assumed that ai-1,ai,ai+1 are not collinearMathworldPlanetmath.

  6. 6.

    A way is said to be simple if it does not meet itself. To be precise, (i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals meet in any point.

  7. 7.

    A polygon is a way a1a2an for which a1=an. Notice that there is no assumptionPlanetmathPlanetmath that the points are coplanarMathworldPlanetmath.

  8. 8.

    A simple polygon is polygon for which the way is simple.

  9. 9.

    A region is a set of points not all collinear, any two of which can be joined by points of a way using only points of the region.

  10. 10.

    A region R is convex if for each pair of points a,bR the open interval (a,b) is contained in R.

  11. 11.

    Let X and Y be two sets of points. If there is a set of points S such that every way joining a point of X to a point of Y meets S then S is said to separate X from Y.

  12. 12.

    If a1a2an is a polygon, then the angles of the polygon are ana1a2,a1a2a3, and so on.

Now assume that all points of the geometry are in one plane. Let P be a polygon. (P is called a plane polygon.)

  1. 1.

    A ray or line which does not go through a vertex of P will be called suitable.

  2. 2.

    An inside point a of P is one for which a suitable ray from a meets P an odd numberMathworldPlanetmathPlanetmath of times. Points that are not on or inside P are said to be outside P.

  3. 3.

    Let {Pi} be a set of polygons. We say that {Pi} dissect P if the following three conditions are satisfied: (i) Pi and Pj do not have a common inside point for ij, (ii) each inside point of P is inside or on some Pi and (iii) each inside point of Pi is inside P.

  4. 4.

    A convex polygon is one whose inside points are all on the same side of any side-line of the polygon.

2 Theorems

Assume that all points are in one plane. Let P be a polygon.

  1. 1.

    It can be shown that P separates the other points of the plane into at least two regions and that if P is simple there are exactly two regions. Moise proves this directly in [3], pp. 16-18.

  2. 2.

    It can be shown that P can be dissected into trianglesMathworldPlanetmath {Ti} such that every vertex of a Ti is a vertex of P.

  3. 3.

    The following theoremMathworldPlanetmath of Euler can be shown: Suppose P is dissected into f>1 polygons and that the total number of vertices of these polygons is v, and the number of open intervals which are sides is e. Then



A plane simple polygon with n sides is called an n-gon, although for small n there are more traditional names:

Number of sides Name of the polygon
3 triangle
4 quadrilateralMathworldPlanetmath
5 pentagonMathworldPlanetmath
6 hexagonMathworldPlanetmath
7 heptagon
8 octagon
10 decagon

A plane simple polygon is also called a Jordan polygon.


  • 1 K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Company, 1960.
  • 2 H.G. Forder, The Foundations of Euclidean Geometry, Dover Publications, 1958.
  • 3 E.E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, 1977.
Title polygon
Canonical name Polygon
Date of creation 2013-03-22 12:10:15
Last modified on 2013-03-22 12:10:15
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 43
Author Mathprof (13753)
Entry type Definition
Classification msc 51-00
Classification msc 51G05
Related topic RegularPolygon
Related topic Semiperimeter
Related topic EquilateralPolygon
Related topic EquiangularPolygon
Related topic Pentagon
Related topic BasicPolygon
Related topic Hexagon
Related topic GeneralizedPythagoreanTheorem
Defines side
Defines vertex
Defines vertices
Defines simple polygon
Defines side-lines
Defines ray
Defines simple way
Defines way
Defines region
Defines convex region
Defines Jordan polygon
Defines angles of a polygon
Defines plane polygon
Defines broken line