parallelogram
A parallelogram (from ancient Greek παραλληλόγραμμος parallelógrammos "bounded by two pairs of parallels") or rhomboid (diamondlike) is a convex, flat square with opposite sides parallel .
Parallelograms are special trapezoids and twodimensional parallelepipeds . Rectangle , diamond (rhombus) and square are special cases of the parallelogram.
properties
A square is a parallelogram if and only if one of the following conditions is met:
 Opposite sides are of the same length and no two opposite sides intersect (no overturned square, socalled antiparallelogram ).
 Two sides are parallel and of equal length.
 Opposite angles are the same size.
 Any two adjacent angles add up to 180 °.
 The diagonals bisect each other.
 The sum of the areas of the squares over the four sides is equal to the sum of the areas of the squares over the two diagonals ( parallelogram equation ).
 It is point symmetric (twofold rotationally symmetric ).
The following applies to every parallelogram:
 Each diagonal divides it into two triangles that are congruent in the same direction .
 Its center of symmetry is the intersection of the diagonals .
 The centers of the squares erected over its sides form a square ( theorem of ThébaultYaglom ).
All parallelograms that have at least one axis of symmetry are rectangles or diamonds .
Formulas
Mathematical formulas for the parallelogram  

Area 


scope  
Interior angle  
height  
Length of the diagonal
(see cosine law ) 

Interior angle  
Parallelogram equation 
Proof of the area formula for a parallelogram
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 .
Generalizations
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 threedimensional 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, truetoparallel mounting, the socalled parallelogram guide, can be created by four joints . Examples:
Switching parallelogram of a derailleur system
Parallel wipers
literature
 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 9783519023562 .
 Wilhelm Killing: Textbook of Analytical Geometry. Part 2, Outlook Verlagsgesellschaft, Bremen 2011, ISBN 9783864035401 .
Web links
 Eric W. Weisstein : Parallelogram . In: MathWorld (English).
 Area and circumference calculation of general and special parallelograms. ( Memento from January 11, 2015 in the Internet Archive ). Retrieved November 18, 2016.
 Introduction to the subject of the parallelogram. (PDF; 920 kB). Retrieved November 18, 2016.
 Parallelogram and diamond. ( Memento from November 19, 2016 in the Internet Archive ; PDF; 225 kB). Retrieved November 18, 2016.