A simple polygon that is not convex 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 of a concave polygon lie partly or wholly outside the polygon. Some

sidelines The "sidelines" are the white or colored lines which mark the outer boundaries of a sports field, running parallel to each other and perpendicular to the goal lines. The sidelines are also where the coaching staff and players out of play ope ...

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 angles of a concave polygon is ×(''n'' − 2) radians, 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 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 can never be concave, but there exist concave polygons with ''n'' sides for any ''n'' > 3. An example of a concave quadrilateral is the 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 or 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