Diagonaalia

Diagonaalia is the partitive singular form of the Finnish noun diagonaali, meaning diagonal. In Finnish, diagonaali describes a straight line that connects two non-adjacent points in a geometric figure, and diagonaalia is used when the diagonal is the object of the sentence. The term also appears in broader mathematical contexts where diagonal-related discussion occurs.

In geometry, a diagonal is a line segment joining two non-consecutive vertices of a polygon or connecting opposite corners in a polyhedron. Diagonals lie inside the figure for convex polygons. For a polygon with n sides, the number of diagonals is n(n−3)/2. For a quadrilateral there are two diagonals.

Properties of diagonals depend on the shape. In a rectangle, the diagonals are equal in length and bisect each other. In a square, the diagonals are not only equal but also perpendicular. Diagonals play a central role in proofs and constructions, such as partitioning polygons into triangles or determining symmetry.

In linear algebra, diagonal refers to the elements aii of a square matrix. A diagonal matrix has nonzero elements only on the diagonal, with all off-diagonal elements equal to zero. The diagonal structure often simplifies many operations, including calculations of powers and determinants for special matrices.

See also: diagonal (geometry), diagonal matrix, polygon, Pythagorean theorem.