Convex and concave polygons

An example of a convex polygon: a regular pentagon

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

Convex polygons

A convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity:

• Every internal angle is less than or equal to 180 degrees.^{[citation needed]}
• Every line segment between two vertices remains inside or on the boundary of the polygon.^{[citation needed]}

A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees.^{[citation needed]} 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.^{[citation needed]}

Every nondegenerate triangle is strictly convex.

Concave or non-convex polygons

An example of a concave polygon.

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 for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985).

See also

• Convex hull

External links

• Weisstein, Eric W., "Convex polygon" from MathWorld.
• Weisstein, Eric W., "Concave polygon" from MathWorld.
• http://www.rustycode.com/tutorials/convex.html