Polygons


In Geometry, a simple polygon is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pair-wise to form a closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general.

The definition given above ensures the following properties:

  • A polygon encloses a region (called its interior) which always has a measurable area.
  • The line segments that make-up a polygon (called sides or edges) meet only at their endpoints, called vertices (singular: vertex) or less formally "corners".
  • Exactly two edges meet at each vertex.
  • The number of edges always equals the number of vertices.

Two edges meeting at a corner are usually required to form an angle that is not straight (180°). Otherwise, the collinear line segments will be considered parts of a single side.

Mathematicians typically use "polygon" to refer only to the shape made up by the line segments, not the enclosed region. However some may use "polygon" to refer to a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). According to the definition in use, this boundary may or may not form part of the polygon itself.

Simple polygons are also called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it. A polygon in the plane is simple if and only if it is topologically equivalent to a circle. Its interior is topologically equivalent to a disk.