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Convex polygon
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a polygon can be either convex or concave.

>convex polygon is a simple polygon
Simple polygon

In geometry, a simple polygon is closed polygonal chain of line segments that do not cross each other. That is, it consists of finitely many line segments, each line segment endpoint is shared by two segments, and the segments do not otherwise intersect....
 whose interior is a convex set
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
. The following properties of a simple polygon are all equivalent to convexity:

A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees.






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Pentagon
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a polygon can be either convex or concave.

Convex polygons

A convex polygon is a simple polygon
Simple polygon

In geometry, a simple polygon is closed polygonal chain of line segments that do not cross each other. That is, it consists of finitely many line segments, each line segment endpoint is shared by two segments, and the segments do not otherwise intersect....
 whose interior is a convex set
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
. The following properties of a simple polygon are all equivalent to convexity:
  • Every internal angle
    Internal angle

    In geometry, an interior angle is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon....
     is less than 180 degrees
    Degree (angle)

    A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
    .
  • Every line segment
    Line segment

    In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
     between two vertices
    Vertex (geometry)

    In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
     remains inside or on the boundary of the polygon.


A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees. 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.

Every nondegenerate triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 is strictly convex.

Concave polygons

A polygon that is not convex is called concave or reentrant. A concave polygon will always have an interior angle with a measure that is greater than 180 degrees.

It is 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 .