A polygon diagonal is a line segment connecting two nonadjacent polygon vertices of a polygon.
For 
,
the number of ways a fixed convex polygon with
vertices
can be divided into triangles by nonintersecting diagonals
is 
,
with 
diagonals in each triangulation, where 
is a Catalan number. This
is Euler's polygon division problem.
Counting the number of regions determined by drawing the diagonals of a regular
polygon with 
vertices is a more difficult problem, as is determining
the number of 
-tuples
of concurrent diagonals (Kok 1972).
The number of regions into which the diagonals of a convex polygon with 
vertices divide its interior,
when no three diagonals are concurrent there, is
 |  |  |
(1)
|
 |  |  |
(2)
|
The first few values, for 
, 4, ..., are 1, 4, 11, 25, 50, 91, 154, 246, ... (OEIS A006522).
