In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

 can be either convex or concave (non-convex).

Convex polygons

A convex polygon is a simple polygon

Simple polygon

In geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....

 whose interior

Interior (topology)

In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....

 is a convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

. The following properties of a simple polygon are all equivalent to convexity:

• Every internal angle
  Internal angle
  In geometry, an interior angle is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon...
  
   is less than or equal to 180 degrees
  Degree (angle)
  A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
  
  .
• Every line segment
  Line segment
  In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
  
   between two vertices
  Vertex (geometry)
  In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
  
   remains inside or on the boundary of the polygon.

A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees. Equivalently, a polygon is strictly convex if every line segment between two nonadjacent vertices of the polygon is strictly interior to the polygon except at its endpoints.

Every nondegenerate triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 is strictly convex.

Concave or non-convex polygons

A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have an interior angle with a measure that is greater than 180 degrees.

It is always possible to cut a concave polygon into a set of convex polygons. A polynomial-time algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

for finding a decomposition into as few convex polygons as possible is described by .