Overview. In everyday language a diagonal is a slanted or oblique straight line. In formal mathematics it most commonly denotes a segment connecting two nonadjacent vertices of a polygon or polyhedron. The idea also appears in linear algebra, engineering and logic with related but distinct meanings.

Definition in geometry

In geometry a diagonal of a polygon is any line segment joining two vertices that are not next to each other along the boundary. For polyhedra one distinguishes face diagonals (across a single face) and space or body diagonals (joining opposite vertices through the interior of the solid).

Counting diagonals

For an n-sided polygon (a polygon) the number of distinct diagonals is given by the formula n(n−3)/2. This counts all segments between pairs of vertices except the n edges. Examples: a square (n=4) has 2 diagonals, a pentagon (n=5) has 5, and a hexagon (n=6) has 9.

Common properties

  • In rectangles the two diagonals are equal in length; in parallelograms they bisect each other.
  • In a square diagonals are equal, bisect each other at right angles, and also bisect the vertex angles.
  • In a rhombus diagonals are perpendicular and bisect the interior angles, but are not generally equal.
  • In polyhedra, body diagonals connect opposite vertices and their lengths relate to the solid's edge lengths (for example, a cuboid has four space diagonals).

Other meanings and applications

In linear algebra the term "main diagonal" denotes entries a_ii of a matrix. The trace of a square matrix is the sum of those diagonal elements. In engineering and architecture diagonal bracing increases stability by resisting shear. The term also appears metaphorically in logic and set theory (for example, Cantor's diagonal argument) and in computer graphics when describing diagonal movement or sampling.

History and notable facts

The word comes via Latin from the Greek diagonios, literally "from angle to angle." Diagonals are a basic tool in proofs, constructions, and computations across mathematics. Their simple definition hides many useful consequences: counting formulas lead to combinatorial reasoning, while diagonal elements in matrices capture invariant sums used throughout linear algebra.