Editing Parallelogram

{{Mergefrom|Rhomboid|date=March 2025}}
{{Short description|Quadrilateral with two pairs of parallel sides}}
{{About|the quadrilateral|the music album|Parallelograms (album)}}
{{Infobox polygon
| name        = Parallelogram 
| image       = Parallelogram.svg
| caption     = This parallelogram is a [[rhomboid]] as it has unequal sides and no right angles.
| type        = [[Quadrilateral]], [[Trapezoid|Trapezium]]
| edges       = 4
| symmetry    = [[Point reflection|C<sub>2</sub>]], [2]<sup>+</sup>, 
| area        = ''bh'' (base × height);<br>''ab'' sin ''θ'' (product of adjacent sides and sine of the vertex angle determined by them)
| properties  = [[Convex polygon]]}}
In [[Euclidean geometry]], a '''parallelogram''' is a [[simple polygon|simple]] (non-[[list of self-intersecting polygons|self-intersecting]]) [[quadrilateral]] with two pairs of [[Parallel (geometry)|parallel]] sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The [[congruence (geometry)|congruence]] of opposite sides and opposite angles is a direct consequence of the Euclidean [[parallel postulate]] and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

By comparison, a quadrilateral with at least one pair of parallel sides is a [[trapezoid]] in American English or a trapezium in British English.

The three-dimensional counterpart of a parallelogram is a [[parallelepiped]].

The word "parallelogram" comes from the Greek παραλληλό-γραμμον, ''parallēló-grammon'', which means "a shape of parallel lines".

==Special cases==
*[[Rectangle]] – A parallelogram with four right angles.
*[[Rhombus]] – A parallelogram with four sides of equal length. Any parallelogram that is a rectangle or a rhombus was traditionally called a [[rhomboid]] but this term is not used in modern mathematics.<ref>{{cite web|url=http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf|title=CIMT - Page no longer available at Plymouth University servers|website=www.cimt.plymouth.ac.uk|url-status=dead|archive-url=https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf|archive-date=2014-05-14}}</ref>
*[[Square (geometry)|Square]] – A parallelogram with four sides of equal length and four right angles.

==Characterizations==
A [[simple polygon|simple]] (non-self-intersecting) [[quadrilateral]] is a parallelogram [[if and only if]] any one of the following statements is true:<ref>Owen Byer, Felix Lazebnik and [[Deirdre Smeltzer]], ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, pp. 51-52.</ref><ref>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.</ref>
*Two pairs of opposite sides are parallel (by definition).
*Two pairs of opposite sides are equal in length.
*Two pairs of opposite angles are equal in measure.
*The [[diagonal]]s bisect each other.
*One pair of opposite sides is [[Parallel (geometry)|parallel]] and equal in length.
*[[Adjacent angles]] are [[supplementary angles|supplementary]].
*Each diagonal divides the quadrilateral into two [[congruence (geometry)|congruent]] [[triangle]]s.
*The sum of the [[Square number|square]]s of the sides equals the sum of the squares of the diagonals. (This is the [[parallelogram law]].)
*It has [[rotational symmetry]] of order 2.
*The sum of the distances from any interior point to the sides is independent of the location of the point.<ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref> (This is an extension of [[Viviani's theorem]].)
*There is a point ''X'' in the plane of the quadrilateral with the property that every straight line through ''X'' divides the quadrilateral into two regions of equal area.<ref>Problem 5, ''2006 British Mathematical Olympiad'', [https://artofproblemsolving.com/community/c6h63970p381087].</ref>
Thus, all parallelograms have all the properties listed above, and [[Converse (logic)|conversely]], if just any one of these statements is true in a simple quadrilateral, then it is considered a parallelogram.

==Other properties==

*Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
*The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
*The area of a parallelogram is also equal to the magnitude of the [[vector cross product]] of two [[adjacent side (polygon)|adjacent]] sides.
*Any line through the midpoint of a parallelogram bisects the area.<ref>Dunn, J.A., and J.E. Pretty, "Halving a triangle", ''Mathematical Gazette'' 56, May 1972, p. 105.</ref>
*Any non-degenerate [[affine transformation]] takes a parallelogram to another parallelogram.
*A parallelogram has [[rotational symmetry]] of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of [[reflectional symmetry]] then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a [[square]].
*The perimeter of a parallelogram is 2(''a'' + ''b'') where ''a'' and ''b'' are the lengths of adjacent sides.
*Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.<ref>{{cite web|last=Weisstein|first=Eric W|title=Triangle Circumscribing|url=http://mathworld.wolfram.com/TriangleCircumscribing.html|work=Wolfram Math World}}</ref>
*The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.<ref name=Weisstein>Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Parallelogram.html</ref>
*If two lines parallel to sides of a parallelogram are constructed [[concurrent lines|concurrent]] to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area.<ref name=Weisstein/>
*The diagonals of a parallelogram divide it into four triangles of equal area.

==Area formula{{anchor|Area}}==
[[File:ParallelogramArea.svg|thumb|180px|alt=A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle|A parallelogram can be rearranged into a rectangle with the same area.]]
[[File:Parallelogram area animated.gif|thumb|180px|Animation for the area formula <math>K = bh</math>.]]
All of the [[Quadrilateral#Area of a convex quadrilateral|area formulas for general convex quadrilaterals]] apply to parallelograms. Further formulas are specific to parallelograms:

A parallelogram with base ''b'' and height ''h'' can be divided into a [[trapezoid]] and a [[right triangle]], and rearranged into a [[rectangle]], as shown in the figure to the left. This means that the [[area]] of a parallelogram is the same as that of a rectangle with the same base and height:

:<math>K = bh.</math>

[[File:Parallelogram area.svg|thumb|The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram]]
The base × height area formula can also be derived using the figure to the right. The area ''K'' of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

:<math>K_\text{rect} = (B+A) \times H\,</math>

and the area of a single triangle is

:<math>K_\text{tri} = \frac{A}{2} \times H. \,</math>

Therefore, the area of the parallelogram is

:<math>K = K_\text{rect} - 2 \times K_\text{tri} = ( (B+A) \times H) - ( A \times H) = B \times H.</math>

Another area formula, for two sides ''B'' and ''C'' and angle θ, is

:<math>K = B \cdot C \cdot \sin \theta.\,</math>

Provided that the parallelogram is a rhombus, the area can be expressed using sides ''B'' and ''C'' and angle <math>\gamma</math> at the intersection of the diagonals:<ref>Mitchell, Douglas W., "The area of a quadrilateral", ''Mathematical Gazette'', July 2009.</ref>

:<math>K = \frac{|\tan \gamma|}{2} \cdot \left| B^2 - C^2 \right|.</math>


When the parallelogram is specified from the lengths ''B'' and ''C'' of two adjacent sides together with the length ''D''<sub>1</sub> of either diagonal, then the area can be found from [[Heron's formula]]. Specifically it is

:<math>K=2\sqrt{S(S-B)(S-C)(S-D_1)}=\frac{1}{2}\sqrt{(B+C+D_1)(-B+C+D_1)(B-C+D_1)(B+C-D_1)},</math>

where <math>S=(B+C+D_1)/2</math> and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into ''two'' congruent triangles.

=== From vertex coordinates ===
Let vectors <math>\mathbf{a},\mathbf{b}\in\R^2</math> and let <math>V = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \end{bmatrix} \in\R^{2 \times 2}</math> denote the matrix with elements of '''a''' and '''b'''. Then the area of the parallelogram generated by '''a''' and '''b''' is equal to <math>|\det(V)| = |a_1b_2 - a_2b_1|\,</math>.

Let vectors <math>\mathbf{a},\mathbf{b}\in\R^n</math> and let <math>V = \begin{bmatrix} a_1 & a_2 & \dots & a_n \\ b_1 & b_2 & \dots & b_n \end{bmatrix} \in\R^{2 \times n}</math>. Then the area of the parallelogram generated by '''a''' and '''b''' is equal to <math>\sqrt{\det(V V^\mathrm{T})}</math>.

Let points <math>a,b,c\in\R^2</math>. Then the [[signed area]] of the parallelogram with vertices at ''a'', ''b'' and ''c'' is equivalent to the determinant of a matrix built using ''a'', ''b'' and ''c'' as rows with the last column padded using ones as follows:
:<math>K = \left| \begin{matrix}
        a_1 & a_2 & 1 \\
        b_1 & b_2 & 1 \\
        c_1 & c_2 & 1
 \end{matrix} \right|. </math>

==Proof that diagonals bisect each other==
[[File:Parallelogram1.svg|right|Parallelogram ABCD]]
To prove that the diagonals of a parallelogram bisect each other, we will use [[congruence (geometry)|congruent]] [[Triangle#Basic facts|triangle]]s:
:<math>\angle ABE \cong \angle CDE</math> ''(alternate interior angles are equal in measure)''
:<math>\angle BAE \cong \angle DCE</math>  ''(alternate interior angles are equal in measure)''.

(since these are angles that a transversal makes with [[Parallel (geometry)|parallel lines]] ''AB'' and ''DC'').

Also, side ''AB'' is equal in length to side ''DC'', since opposite sides of a parallelogram are equal in length.

Therefore, triangles ''ABE'' and ''CDE'' are congruent (ASA postulate, ''two corresponding angles and the included side'').

Therefore,
:<math>AE = CE</math>
:<math>BE = DE.</math>

Since the diagonals ''AC'' and ''BD'' divide each other into segments of equal length, the diagonals bisect each other.

Separately, since the diagonals ''AC'' and ''BD'' bisect each other at point ''E'', point ''E'' is the midpoint of each diagonal.

== Lattice of parallelograms==
Parallelograms can tile the plane by translation. If edges are equal, or angles are right, the symmetry of the lattice is higher. These represent the four [[Bravais_lattice#In_2_dimensions|Bravais lattices in 2 dimensions]].
{| class=wikitable
|+ Lattices
|-
!Form||Square||Rectangle||Rhombus||Rhomboid
|-
!System
!Square<BR>(tetragonal)
!Rectangular<BR>(orthorhombic)
!Centered rectangular<BR>(orthorhombic)
!Oblique<BR>(monoclinic)
|- align=center
!Constraints
|α=90°, a=b
|α=90°
|a=b
|None
|- align=center
![[List_of_planar_symmetry_groups#Wallpaper_groups|Symmetry]]
|p4m, [4,4], order 8''n''||colspan=2|pmm, [∞,2,∞], order 4''n''||p1, [∞<sup>+</sup>,2,∞<sup>+</sup>], order 2''n''
|- align=center
!Form
|[[File:Lattice of squares.svg|160px]]
|[[File:Lattice of rectangles.svg|160px]]
|[[File:Lattice of rhombuses.svg|160px]]
|[[File:Lattice of rhomboids.svg|160px]]
|}

==Parallelograms arising from other figures==

===Automedian triangle===
An [[automedian triangle]] is one whose [[median (geometry)|medians]] are in the same proportions as its sides (though in a different order). If ''ABC'' is an automedian triangle in which vertex ''A'' stands opposite the side ''a'',  ''G'' is the [[centroid]] (where the three medians of ''ABC'' intersect), and ''AL'' is one of the extended medians of ''ABC'' with ''L'' lying on the circumcircle of ''ABC'', then ''BGCL'' is a parallelogram.

===Varignon parallelogram===
{{main|Varignon's theorem}}

[[File:varignon_parallelogram.svg|thumb|Proof without words of Varignon's theorem ]]

[[Varignon's theorem]] holds that the [[midpoint]]s of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its ''Varignon parallelogram''. If the quadrilateral is [[Convex polygon|convex]] or [[Concave polygon|concave]] (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.

[[Proof without words]] (see figure):

# An arbitrary quadrilateral and its diagonals.
# Bases of similar triangles are parallel to the blue diagonal.
# Ditto for the red diagonal.
# The base pairs form a parallelogram with half the area of the quadrilateral, ''A<sub>q</sub>'', as the sum of the areas of the four large triangles, ''A<sub>l</sub>'' is 2 ''A<sub>q</sub>'' (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, ''A<sub>s</sub>'' is a quarter of ''A<sub>l</sub>'' (half linear dimensions yields quarter area), and the area of the parallelogram is ''A<sub>q</sub>'' minus ''A<sub>s</sub>''.

===Tangent parallelogram of an ellipse===

For an [[ellipse]], two diameters are said to be [[Conjugate diameters|conjugate]] if and only if the [[tangent line]] to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding [[tangent parallelogram]], sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area.

It is possible to [[Compass and straightedge constructions|reconstruct]] an ellipse from any pair of conjugate diameters, or from any tangent parallelogram.

===Faces of a parallelepiped===

A [[parallelepiped]] is a three-dimensional figure whose six [[face (geometry)|faces]] are parallelograms.

==See also==
* [[parallelogon]], generalisation encompassing hexagons as well as quadrilaterals
* [[zonogon]], generalisation to polygons with any even number of sides
* [[Antiparallelogram]]
* [[Levi-Civita parallelogramoid]]

==References==
{{reflist}}

==External links==
{{Commons category|Parallelograms}}
*[http://www.elsy.at/kurse/index.php?kurs=Parallelogram+and+Rhombus&status=public Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)]
*{{MathWorld |urlname=Parallelogram |title=Parallelogram}}
*[http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/index.php Interactive Parallelogram --sides, angles and slope]
*[http://www.cut-the-knot.org/Curriculum/Geometry/AreaOfParallelogram.shtml Area of Parallelogram] at [[cut-the-knot]]
*[http://www.cut-the-knot.org/Curriculum/Geometry/EquiTriOnPara.shtml Equilateral Triangles On Sides of a Parallelogram] at [[cut-the-knot]]
*[http://www.mathopenref.com/parallelogram.html Definition and properties of a parallelogram] with animated applet
*[http://www.mathopenref.com/parallelogramarea.html Interactive applet showing parallelogram area calculation] interactive applet

{{Polygons}}
[[Category:Types of quadrilaterals]]
[[Category:Elementary shapes]]