Convex and concave polygons

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An example of a convex polygon: a regular pentagon

In geometry, a polygon can be either convex or concave (non-convex or reentrant).

Convex polygons[edit]

A convex polygon is a simple polygon whose interior is a convex set.[1] The following properties of a simple polygon are both equivalent to convexity:


Every nondegenerate triangle is strictly convex.

Concave or non-convex polygons[edit]

An example of a concave polygon.

A simple polygon that is not convex is called concave,[2] non-convex[3] or reentrant.[4] A concave polygon will always have an interior angle with a measure that is greater than 180 degrees.[5]

It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985).[6]

See also[edit]


  1. ^ Definition and properties of convex polygons with interactive animation.
  2. ^ McConnell, Jeffrey J. (2006), Computer Graphics: Theory Into Practice, p. 130, ISBN 0-7637-2250-2 .
  3. ^ Leff, Lawrence (2008), Let's Review: Geometry, Hauppauge, NY: Barron's Educational Series, p. 66, ISBN 978-0-7641-4069-3 
  4. ^ Mason, J.I. (1946), "On the angles of a polygon", The Mathematical Gazette (The Mathematical Association) 30 (291): 237–238, JSTOR 3611229 .
  5. ^ Definition and properties of concave polygons with interactive animation.
  6. ^ Chazelle, Bernard; Dobkin, David P. (1985), "Optimal convex decompositions", in Toussaint, G.T., Computational Geometry, Elsevier, pp. 63–133 .

External links[edit]