simple polygon In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...

that is not

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.

Polygon

Some lines containing interior points of a concave polygon intersect its boundary at more than two points. Some

diagonals In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

of a concave polygon lie partly or wholly outside the polygon. Some sidelines of a concave polygon fail to divide the plane into two half-planes one of which entirely contains the polygon. None of these three statements holds for a convex polygon. As with any simple polygon, the sum of the

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

s of a concave polygon is ×(''n'' − 2)

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...

s, equivalently 180×(''n'' − 2) degrees (°), where ''n'' is the number of sides. It is always possible to

partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...

a concave polygon into a set of convex polygons. A

polynomial-time algorithm In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...

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

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

can never be concave, but there exist concave polygons with ''n'' sides for any ''n'' > 3. An example of a concave

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

is the

dart Dart or DART may refer to: * Dart, the equipment in the game of darts Arts, entertainment and media * Dart (comics), an Image Comics superhero * Dart, a character from ''G.I. Joe'' * Dart, a ''Thomas & Friends'' railway engine character * Dar ...

. At least one interior angle does not contain all other vertices in its edges and interior. The convex hull of the concave polygon's vertices, and that of its edges, contains points that are exterior to the polygon.

Notes

External links

*{{mathworld , urlname=ConcavePolygon , title=Concave polygon Types of polygons