Diagonal
GeometryA 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 from one corner to another corner that is not right next to it. It is not a side - it cuts across the inside of the shape.
Example
A square has $2$ diagonals that cross in the middle. A rectangle's diagonal cuts it into two triangles. You can see the diagonals when you draw an X from corner to corner inside a square.
Key Insight
Diagonals help you measure across a shape. The diagonal of a rectangle is longer than either side - it is the longest distance inside the rectangle. TV screen sizes are measured as diagonals.
Definition
A diagonal of a polygon is a line segment connecting two non-adjacent vertices. An $n$-gon has $n(n-3)/2$ diagonals. Diagonals divide polygons into triangles (triangulation). In special quadrilaterals: a square's diagonals are equal, perpendicular, and bisect each other; a rhombus' diagonals are perpendicular; a rectangle's diagonals are equal.
Example
Pentagon ($5$ sides): diagonals $= 5(5-3)/2 = 5$. Hexagon: $6(6-3)/2 = 9$. For a square with side $s$: diagonal $= s\sqrt{2}$. For a $3$-$4$-$5$ right triangle (degenerate polygon), the "diagonal" is the hypotenuse $= 5$.
Key Insight
The formula $n(n-3)/2$ comes from: choose any $2$ of $n$ vertices (giving $C(n,2) = n(n-1)/2$ segments), subtract the $n$ sides, leaving $n(n-1)/2 - n = n(n-3)/2$. This combinatorial counting connects geometry to combinations.
Definition
The diagonals of a convex polygon partition its interior into triangles (triangulation). An $n$-gon can be triangulated into $(n-2)$ triangles. For a convex polygon, all diagonals lie inside; for concave polygons, some diagonals may pass outside. The diagonal count $n(n-3)/2$ grows as $O(n^2)$, and for large $n$, the triangulation count grows as the Catalan number $C(n-2)$.
Example
Hexagon triangulated from one vertex: $4$ diagonals from one vertex, creating $4$ triangles ($n-2 = 4$). Total triangulation count for a convex hexagon = Catalan number $C(4) = 14$ distinct triangulations.
Key Insight
The number of triangulations of a convex $n$-gon equals the Catalan number $C(n-2)$. Catalan numbers appear across combinatorics (parenthesizations, binary trees, ballot problems). This connection between polygon triangulations and Catalan numbers illustrates how a geometric question can open a vast combinatorial landscape.