A triangulation of a polygon is a set of T chords that divide the polygon into…

Result: A triangulation of an n-vertex convex polygon has n-3 chords (diagonals) and n-2 triangles.

• Proof by induction:

  Base case: For n = 3 (a triangle) there are 0 chords and 1 triangle, which matches n-3 = 0 and n-2 = 1.

  Induction step: Any convex polygon has an ear (a triangle formed by two polygon edges and one chord). Removing that ear yields a convex polygon with n-1 vertices; the removal reduces the count of chords by 1 and the count of triangles by 1. If for n-1 vertices the counts are (n-1)-3 and (n-1)-2, then for n vertices they are ((n-1)-3)+1 = n-3 chords and ((n-1)-2)+1 = n-2 triangles. Thus the formulas hold for all n ≥ 3.

• Alternative counting argument:

  Let T be the number of triangles and C the number of chords. Every triangle has 3 edges, and counting all triangle edges counts the polygon boundary edges once (n of them) and each internal chord twice. Therefore 3T = n + 2C.

  Also each chord adds exactly one triangle compared to no chords, so T = C + 1. Solving the two equations gives T = n-2 and C = n-3.