simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...

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 � ...

of a concave polygon lie partly or wholly outside the polygon. Some

sidelines In sports, out of bounds (or out-of-bounds) refers to being outside the playing boundaries of the Pitch (sports field), field. The legality of going out of bounds (intentionally or not), and the ease of prevention, vary by sport. Sports that use ...

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 adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...

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. It is defined such that one radian is the angle subtended at ...

s, equivalently 180°(''n'' − 2) degrees, where ''n'' is the number of sides. It is always possible to partition a concave polygon into a set of convex polygons. A

polynomial-time algorithm In theoretical 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 p ...

for finding a decomposition into as few convex polygons as possible is described by .. According to Euclidean geometry, a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

is the

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

. At least one interior angle does not contain all other vertices in its edges and interior. The

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

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