Vierecks

Viereck is the German term for a four‑sided polygon, commonly called a quadrilateral in English. It has four vertices and four sides, and the sum of its interior angles is always 360 degrees. The standard plural is Vierecke, while Vierecks is the genitive singular form (des Vierecks), used to indicate possession.

Vierecke can be classified in several ways. By shape, they include parallelograms (opposite sides are parallel), rectangles (all angles are right), squares (all sides and all angles are right), rhombuses (all sides are equal), and trapezoids (at least one pair of opposite sides is parallel). By angle, they can be convex (no interior angle exceeds 180 degrees) or concave. A cyclic quadrilateral has all four vertices on a common circle, and many results simplify in this case.

Key properties and formulas include: the area can be found by dividing the figure into two triangles or, for a general quadrilateral, by the formula involving diagonals: area = 1/2 × d1 × d2 × sin(theta), where d1 and d2 are the lengths of the diagonals and theta is the angle between them. For a rectangle, area = base × height; for a parallelogram, area = base × height. Brahmagupta’s formula gives the area of a cyclic quadrilateral: area = sqrt((s−a)(s−b)(s−c)(s−d)) with s the semiperimeter. Ptolemy’s theorem applies to cyclic quadrilaterals: the product of the diagonals equals the sum of the products of opposite sides.

Vierecks appear across geometry, design, architecture, and computer graphics, serving as a foundational shape for both theory and application.