Diagonal
A diagonal is a line segment connecting two non-adjacent vertices of a polygon.
Formula
\text{Number of diagonals} = n(n-3)/2 \text{ (for an } n\text{-gon)}
Definition
A diagonal is a straight line drawn inside a polygon connecting one corner to another that is not right next to it, cutting across the interior rather than forming a side. An $n$-gon has $n(n-3)/2$ diagonals, obtained by choosing any $2$ of $n$ vertices ($C(n,2)=n(n-1)/2$ segments) and subtracting the $n$ sides. Diagonals partition a convex polygon into triangles (triangulation), an $n$-gon into $(n-2)$ triangles, and the number of distinct triangulations of a convex $n$-gon equals the Catalan number $C(n-2)$.
Example
A square has $2$ diagonals crossing in the middle, and a rectangle's diagonal cuts it into two triangles; TV screen sizes are measured as diagonals. A pentagon has $5(5-3)/2=5$ diagonals, a hexagon $6(6-3)/2=9$; a square's diagonal is $s\sqrt{2}$. Triangulating a hexagon from one vertex uses $4$ diagonals to create $4$ triangles ($n-2$), and the total number of distinct triangulations of a convex hexagon is the Catalan number $C(4)=14$.
Key Insight
Diagonals help measure across a shape, the diagonal of a rectangle is the longest distance inside it. In special quadrilaterals, a square's diagonals are equal, perpendicular, and bisect each other, a rhombus's are perpendicular, and a rectangle's are equal. Catalan numbers, which count polygon triangulations, also appear across combinatorics in parenthesizations, binary trees, and ballot problems, showing how a simple geometric counting question opens onto a vast combinatorial landscape.