parallelogram

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.

properties

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 .

Formulas

Mathematical formulas for the parallelogram
Area ${\ displaystyle A = a \ cdot h_ {a} = b \ cdot h_ {b} = \ left | \ left | {\ overrightarrow {AB}} \ times {\ overrightarrow {AD}} \ right | \ right |}$

${\ displaystyle A = a \ cdot b \ cdot \ sin (\ alpha) = a \ cdot b \ cdot \ sin (\ beta) = {\ frac {e \ cdot f \ cdot \ sin (\ theta)} {2 }}}$
Via transformation into a rectangle with the determinant :
${\ displaystyle A = \ det {\ begin {pmatrix} a_ {x} && b_ {x} \\ a_ {y} && b_ {y} \ end {pmatrix}} = a_ {x} \ cdot b_ {y} -b_ {x} \ cdot a_ {y}}$

scope ${\ displaystyle U = 2 \ times a + 2 \ times b = 2 \ times (a + b)}$
Interior angle ${\ displaystyle \ alpha = \ gamma, \ quad \ beta = \ delta, \ quad \ alpha + \ beta = 180 ^ {\ circ}}$
height ${\ displaystyle h_ {a} = b \ cdot \ sin (\ alpha)}$
${\ displaystyle h_ {b} = a \ cdot \ sin (\ beta)}$
Length of the diagonal

(see cosine law )

${\ displaystyle {\ begin {array} {ccl} e & = {\ sqrt {a ^ {2} + b ^ {2} -2 \ cdot a \ cdot b \ cdot \ cos (\ beta)}} \\ & = {\ sqrt {a ^ {2} + b ^ {2} +2 \ cdot a \ cdot b \ cdot \ cos (\ alpha)}} \ end {array}}}$
${\ displaystyle {\ begin {array} {ccl} f & = {\ sqrt {a ^ {2} + b ^ {2} -2 \ cdot a \ cdot b \ cdot \ cos (\ alpha)}} \\ & = {\ sqrt {a ^ {2} + b ^ {2} +2 \ cdot a \ cdot b \ cdot \ cos (\ beta)}} \ end {array}}}$
Interior angle ${\ displaystyle \ alpha = \ gamma, \ quad \ beta = \ delta, \ quad \ alpha + \ beta = 180 ^ {\ circ}}$
Parallelogram equation ${\ displaystyle e ^ {2} + f ^ {2} = 2 \ cdot (a ^ {2} + b ^ {2})}$

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 .${\ displaystyle b}$ ${\ displaystyle h}$

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: ${\ displaystyle A}$

${\ displaystyle {\ begin {array} {cccl} A & = && (\ color {YellowOrange} a_ {x} \ color {black} + \ color {ForestGreen} b_ {x} \ color {black}) \ cdot (\ color {red} a_ {y} \ color {black} + \ color {blue} b_ {y} \ color {black}) - 2 \ cdot (\ color {YellowOrange} a_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ color {black} / 2 + \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ color {black} + \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {blue} b_ {y} \ color {black} / 2) \\ & = && \ color {YellowOrange} a_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ color {black} + \ color {YellowOrange} a_ {x} \ color {black} \ cdot \ color {blue} b_ {y} \ color {black} + \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ color {black} + \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {blue} b_ {y} \\ && \ color {black} - & \ color {YellowOrange} a_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ color {black} \ quad \ quad \ quad -2 \ cdot \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ color {black} - \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {blue } b_ {y} \\ & = && \ quad \ quad \ quad \ quad \ color {YellowOrange} a_ {x} \ color {black} \ cdot \ color {blue} b_ {y} \ color {black} - \ color {ForestGreen} b_ {x} \ color {black} \ cdot \ color {red} a_ {y} \ end {array}}}$

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 . ${\ displaystyle a}$${\ displaystyle b}$ ${\ displaystyle h_ {a}}$

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.${\ displaystyle a}$${\ displaystyle b}$ ${\ displaystyle h_ {a}}$${\ displaystyle E}$

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 . ${\ displaystyle n}$ ${\ displaystyle \ {\ alpha _ {1} \ cdot p_ {1} + \ alpha _ {2} \ cdot p_ {2} + \ dotsb + \ alpha _ {n} \ cdot p_ {n} \ mid 0 \ leq \ alpha _ {i} \ leq 1 \}}$${\ displaystyle p_ {i}}$${\ displaystyle n}$

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:

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 978-3-519-02356-2 .
• Wilhelm Killing: Textbook of Analytical Geometry. Part 2, Outlook Verlagsgesellschaft, Bremen 2011, ISBN 978-3-86403-540-1 .