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A parallelogram (from ancient Greek παραλληλό-γραμμος paralleló-grammos "bounded by two pairs of parallels") or rhomboid (diamond-like) is a convex, flat square with opposite sides parallel .

Parallelograms are special trapezoids and two-dimensional parallelepipeds . Rectangle , diamond (rhombus) and square are special cases of the parallelogram.


A square is a parallelogram if and only if one of the following conditions is met:

The following applies to every parallelogram:

All parallelograms that have at least one axis of symmetry are rectangles or diamonds .


Mathematical formulas for the parallelogram

Via transformation into a rectangle with the determinant :

Interior angle
Length of the diagonal

(see cosine law )

Interior angle
Parallelogram equation

Proof of the area formula for a parallelogram

Six partial areas are subtracted from the large rectangle
Animation for calculating the area of a parallelogram. The area is equal to the product of the length of a base side and the associated height .

The area of the adjacent black parallelogram can be obtained by subtracting the six small areas with colored edges from the area of ​​the large rectangle . Because of the symmetry and the interchangeability of the multiplication , you can also subtract twice the three small areas below the parallelogram from the large rectangle. So it is:

Construction of a parallelogram

A parallelogram, wherein the side lengths and as well as the height is given, is provided with ruler and compass constructible .

Parallelogram with the given side lengths and and the
height . The point can be freely selected for the construction of the right angle . Animation with a pause of 10 s at the end.


A generalization to dimensions is the parallelotope , explained as the set and its parallel displacements . They are linearly independent vectors . Parallelotopes are point symmetrical .

The three-dimensional parallelotope is the parallelepiped . Its side surfaces are six parallelograms that are congruent in pairs and lie in parallel planes . A parallelepiped has twelve edges, four of which are parallel and of equal length, and eight corners in which these edges converge at a maximum of three different angles .

Use in technology

Parallelograms are often found in mechanics. A movable, true-to-parallel mounting, the so-called parallelogram guide, can be created by four joints . Examples:


  • F. Wolff: Textbook of Geometry. Fourth improved edition, printed and published by G. Reimer, Berlin 1845 ( online copy ).
  • P. Kall: Linear Algebra for Economists. Springer Fachmedien, Wiesbaden 1984, ISBN 978-3-519-02356-2 .
  • Wilhelm Killing: Textbook of Analytical Geometry. Part 2, Outlook Verlagsgesellschaft, Bremen 2011, ISBN 978-3-86403-540-1 .

Web links

Commons : Parallelogram  - collection of images, videos and audio files
Wiktionary: Parallelogram  - explanations of meanings, word origins, synonyms, translations
Wiktionary: Rhomboid  - explanations of meanings, word origins, synonyms, translations