Convex polygon

An example of a convex polygon: a regular pentagon

A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set.[1] In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.

A simple polygon which is not convex is called concave.


The following properties of a simple polygon are all equivalent to convexity:

Additional properties of convex polygons include:

Every polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if not self-intersecting, is convex. However, not every convex polygon can be inscribed in a circle.

Strict convexity

The following properties of a simple polygon are all equivalent to strict convexity:

Every nondegenerate triangle is strictly convex.

See also


  1. Definition and properties of convex polygons with interactive animation.
  2. -, Christos. "Is the area of intersection of convex polygons always convex?". Math Stack Exchange.
  3. Weisstein, Eric W. "Triangle Circumscribing". Wolfram Math World.
  4. Lassak, M. (1993). "Approximation of convex bodies by rectangles". Geometriae Dedicata. 47: 111. doi:10.1007/BF01263495.
  5. Jim Belk. "What's the average width of a convex polygon?". Math Stack Exchange.
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