Triangulation of a convex polygon

A triangulation of a convex polygon divides a convex polygon into triangles using diagonals between its existing vertices of a polytope, with no diagonal crossings in the interior. For $m\ge 3$ vertices it uses $m-3$ diagonals and produces $m-2$ triangles. Fixing one boundary edge, its adjacent triangle separates two smaller polygon regions; summing over the possible third vertices of a polytope gives the Catalan recurrence. Thus the number of triangulations of a convex polygon is $C_{m-2}$, with $C_n$ the Catalan number. A quadrilateral has two choices and a pentagon has five.