Vierecken

Vierecken, or quadrilaterals, are polygons with four sides and four vertices. The interior angles sum to 360 degrees. They can be convex, where all interior angles are less than 180 degrees, or concave, where one interior angle exceeds 180 degrees.

Common types include parallelograms (opposite sides are parallel), rectangles (all angles are 90 degrees), rhombuses (all sides equal), and squares (both). A trapezoid (also called trapezium in some regions) has at least one pair of parallel sides; an isosceles trapezoid has equal base angles. A kite has two distinct pairs of adjacent equal sides. Cyclic quadrilaterals have all four vertices on a circle, and for them opposite angles sum to 180 degrees. There are also tangential quadrilaterals, which have an incircle, and bicentric quadrilaterals, which have both an incircle and a circumcircle.

Diagonals connect opposite vertices and typically intersect inside a convex quadrilateral. The area can be computed as A = 1/2 d1 d2 sin(phi), where d1 and d2 are the diagonal lengths and phi is the angle between them. Alternatively, the area can be found by dividing the figure into two triangles along a diagonal. For cyclic quadrilaterals, Brahmagupta’s formula gives the area in terms of the side lengths and semiperimeter.

Vierecken are central to geometry education and have applications in architecture, computer graphics, surveying, and land plotting, where four-sided figures naturally arise in design and measurement tasks.