raute

Raute is the German word for the geometric figure known in English as a rhombus or lozenge. It describes a four-sided polygon with all sides of equal length and opposite sides parallel. In this sense, a raute is a special kind of parallelogram.

In a raute, the diagonals intersect at right angles and bisect each other, and they also bisect the angles at the vertices. These properties lead to useful area formulas: the area equals half the product of the diagonals, A = (d1 × d2) / 2. If the side length is a and the interior angle is θ, the area can also be written as A = a^2 sin θ.

Special cases include the square: when the interior angles are all right angles, the raute becomes a square.

Raute shapes appear in various design and mathematical contexts. They are common in tiling and lattice patterns, where rhombi tile the plane in rhombille tilings or in lozenge tilings. In art and heraldry, the diamond-shaped motif of the raute is a recurring geometric element.

The term raute is also used as a proper noun in modern usage. It is the name of a Finnish company, Raute Corporation, which specializes in machinery for veneer and plywood production. Thus, while primarily a geometric term, raute also appears in branding and industry names.