An example of a convex polygon: a regular pentagon.

In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting).[1] Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.

A strictly convex polygon is a convex polygon such that no line contains two of its edges. 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.


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 non-degenerate triangle is strictly convex.

See also


  1. ^ Definition and properties of convex polygons with interactive animation.
  2. ^ Chandran, Sharat; Mount, David M. (1992). "A parallel algorithm for enclosed and enclosing triangles". International Journal of Computational Geometry & Applications. 2 (2): 191–214. doi:10.1142/S0218195992000123. MR 1168956.
  3. ^ Weisstein, Eric W. "Triangle Circumscribing". Wolfram Math World.
  4. ^ Lassak, M. (1993). "Approximation of convex bodies by rectangles". Geometriae Dedicata. 47: 111–117. doi:10.1007/BF01263495. S2CID 119508642.
  5. ^ Belk, Jim. "What's the average width of a convex polygon?". Math Stack Exchange.