Parallelogramm

Parallelogramm is a simple quadrilateral with two pairs of opposite sides parallel. Because opposite sides are parallel, they are also equal in length, and opposite angles are equal. The diagonals of a parallelogramm bisect each other.

If a and b are the lengths of two adjacent sides and theta is the interior angle between them, the area A can be computed as A = a b sin(theta). Equivalently A = base × height, where height is the perpendicular distance between the bases.

The diagonals p and q have lengths given by p^2 = a^2 + b^2 + 2ab cos theta and q^2 = a^2 + b^2 − 2ab cos theta. The diagonals intersect at their midpoints. The area can also be expressed as A = (1/2) p q sin(phi) where phi is the angle between the diagonals.

Special parallelograms: a rectangle is a parallelogram with right angles, in which diagonals are equal in length. A rhombus is a parallelogram with equal side lengths, in which diagonals are perpendicular. A square is a parallelogram that is both a rectangle and a rhombus.

In vector form, if the two adjacent side vectors are u and v, the parallelogram consists of points t u + s v with 0 ≤ t, s ≤ 1, and its area is |u × v| (2D: |det(u,v)|).