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).
