a diagonal
/ dy-AG-uh-nul /
In any polygon with at least four corners, you can draw a shortcut from one corner straight across to another corner that is not its neighbour. That cross-link is a diagonal. The two cross-braces of a rectangular gate, or the line on a baseball diamond from home plate to second base, are diagonals — they skip the sides and reach a non-adjacent vertex.
Precisely, a diagonal of a polygon is a line segment joining two vertices that are not connected by a side (that is, two non-adjacent vertices). A triangle has none, since every pair of its three vertices is already joined by a side. The number of diagonals of an n-sided polygon is n(n-3)/2: each vertex connects to n-3 others (all but itself and its two neighbours), giving n(n-3) endpoints, and dividing by 2 avoids counting each diagonal from both ends. So a quadrilateral has 4·1/2 = 2 diagonals, a pentagon 5·2/2 = 5, a hexagon 6·3/2 = 9.
Diagonals are workhorses of plane geometry. Drawing all diagonals from a single vertex of a convex n-gon cuts it into n-2 triangles — exactly the picture that proves the interior-angle sum is (n-2)·180 degrees. In the quadrilateral family the behaviour of the diagonals is the diagnostic feature: a parallelogram's diagonals bisect each other, a rectangle's are equal, a rhombus's are perpendicular, a kite's are perpendicular with one bisecting the other. A caution for concave polygons: a 'diagonal' joining two non-adjacent vertices may pass outside the figure, so 'diagonal' guarantees neither that the segment stays inside nor that it lies wholly within the interior.
A hexagon has 6 vertices. Each vertex links to 6-3 = 3 non-neighbours, so 6·3 = 18 endpoint-uses; dividing by 2 gives 9 diagonals. From one vertex you can draw 3 of them, slicing the hexagon into 6-2 = 4 triangles.
Counting diagonals with n(n-3)/2, and using vertex diagonals to triangulate.
A triangle has zero diagonals — there is no non-adjacent pair of vertices to join. In a concave polygon a diagonal can run outside the figure, so 'diagonal' does not imply 'interior'.