View source for Rhombus

{{Short description|Quadrilateral with sides of equal length}}
{{Other uses}}
{{pp-vandalism|small=yes}}
{{Infobox Polygon
| name       = Rhombus
| image      = rhombus.svg
| caption    = A rhombus in two different orientations
| type       = [[quadrilateral]], [[trapezoid]], [[parallelogram]], [[kite (geometry)|kite]]
| edges      = 4
| symmetry   = [[Dihedral symmetry|Dihedral]] (D<sub>2</sub>), [2], (*22), order 4
| coxeter    = {{CDD|node_1|sum|node_1}}
| schläfli   = {&nbsp;} + {&nbsp;}<BR>{2<sub>&alpha;</sub>}
| area       = <math>K = \frac{p \cdot q}{2} </math> (half the product of the diagonals)
| dual       = [[rectangle]]
| properties = [[convex polygon|convex]], [[isotoxal figure|isotoxal]]}}
[[File:Symmetries of square.svg|280px|thumb|The rhombus has a square as a special case, and is a special case of a [[Kite (geometry)|kite]] and [[parallelogram]].]]

In plane [[Euclidean geometry]], a '''rhombus''' ({{plural form}}: '''rhombi''' or '''rhombuses''') is a [[quadrilateral]] whose four sides all have the same length. Another name is '''equilateral quadrilateral''', since equilateral means that all of its sides are equal in length. The rhombus is often called a "'''diamond'''", after the [[Diamonds (suit)|diamonds]] suit in [[playing card]]s which resembles the projection of an [[Octahedron#Orthogonal projections|octahedral]] [[diamond]], or a [[lozenge (shape)|lozenge]], though the former sometimes refers specifically to a rhombus with a 60° [[angle]] (which some authors call a '''calisson''' after [[calisson|the French sweet]]<ref>{{Cite book|url=https://books.google.com/books?id=2F_0DwAAQBAJ&pg=PA28|title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century|isbn = 9781614442165|last1 = Alsina|first1 = Claudi|last2 = Nelsen|first2 = Roger B.|date = 31 December 2015| publisher=American Mathematical Soc. }}</ref>—also see [[Polyiamond]]), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is [[simple polygon|simple]] (non-self-intersecting), and is a special case of a [[parallelogram]] and a [[Kite (geometry)|kite]]. A rhombus with right angles is a [[square]].<ref>Note: [[Euclid]]'s original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition. See, e.g., {{cite journal
 | last = De Villiers | first = Michael
 | date = February 1994
 | issue = 1
 | journal = [[For the Learning of Mathematics]]
 | jstor = 40248098
 | pages = 11–18
 | title = The role and function of a hierarchical classification of quadrilaterals
 | volume = 14}}</ref>

==Etymology==
The word "rhombus" comes from {{langx|grc|ῥόμβος|rhómbos}}, meaning something that spins,<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dr%28o%2Fmbos {{lang|grc|ῥόμβος}}] {{Webarchive|url=https://web.archive.org/web/20131108114843/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dr%28o%2Fmbos |date=2013-11-08 }}, Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> which derives from the verb {{wikt-lang|grc|ῥέμβω}}, <small>romanized:</small> {{transl|grc|rhémbō}}, meaning "to turn round and round."<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dr%28e%2Fmbw ρέμβω] {{Webarchive|url=https://web.archive.org/web/20131108114840/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dr%28e%2Fmbw |date=2013-11-08 }}, Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> The word was used both by [[Euclid]] and [[Archimedes]], who used the term "solid rhombus" for a [[bicone]], two right circular [[cone]]s sharing a common base.<ref>{{Cite web |url=http://www.pballew.net/rhomb |title=The Origin of Rhombus |access-date=2005-01-25 |archive-url=https://web.archive.org/web/20150402143657/http://www.pballew.net/rhomb |archive-date=2015-04-02 |url-status=usurped }}</ref>

The surface we refer to as ''rhombus'' today is a [[cross section (geometry)|cross section]] of the bicone on a plane through the apexes of the two cones.

==Characterizations==
[[File:Haltern am See, Naturpark Hohe Mark, Hohemarkenbusch, Baumstamm -- 2024 -- 4411 (kreativ 2).jpg|thumb|upright|An [[Intentional camera movement|ICM]] photo with a diamond-shaped composition.]]
A [[simple polygon|simple]] (non-[[list of self-intersecting polygons|self-intersecting]]) quadrilateral is a rhombus [[if and only if]] it is any one of the following:<ref>Zalman Usiskin and Jennifer Griffin, "[https://books.google.com/books?id=ff0nDwAAQBAJ&q=rhombus The Classification of Quadrilaterals. A Study of Definition] {{Webarchive|url=https://web.archive.org/web/20200226195300/https://books.google.com/books?id=ff0nDwAAQBAJ&printsec=frontcover#v=onepage&q=rhombus&f=false |date=2020-02-26 }}", Information Age Publishing, 2008, pp. 55-56.</ref><ref>Owen Byer, Felix Lazebnik and [[Deirdre Smeltzer]], ''[https://books.google.com/books?id=W4acIu4qZvoC&q=rhombus Methods for Euclidean Geometry] {{Webarchive|url=https://web.archive.org/web/20190901191543/https://books.google.com/books?id=W4acIu4qZvoC&printsec=frontcover#v=onepage&q=rhombus&f=false |date=2019-09-01 }}'', Mathematical Association of America, 2010, p. 53.</ref>
*a [[parallelogram]] in which a [[diagonal]] bisects an [[Internal and external angle|interior angle]]
*a parallelogram in which at least two consecutive sides are equal in length
*a parallelogram in which the diagonals are perpendicular (an [[orthodiagonal]] parallelogram)
*a quadrilateral with four sides of equal length (by definition)
*a quadrilateral in which the diagonals are [[perpendicular]] and [[Bisection|bisect]] each other
*a quadrilateral in which each diagonal bisects two opposite interior angles
*a quadrilateral ''ABCD'' possessing a point ''P'' in its plane such that the four triangles ''ABP'', ''BCP'', ''CDP'', and ''DAP'' are all [[Congruence (geometry)|congruent]]<ref>Paris Pamfilos (2016), "A Characterization of the Rhombus", ''[[Forum Geometricorum]]'' '''16''', pp. 331–336, [http://forumgeom.fau.edu/FG2016volume16/FG201640.pdf] {{Webarchive|url=https://web.archive.org/web/20161023135753/http://forumgeom.fau.edu/FG2016volume16/FG201640.pdf|date=2016-10-23}}</ref>
*a quadrilateral ''ABCD'' in which the [[Incircle and excircles of a triangle|incircle]]s in triangles ''ABC'', ''BCD'', ''CDA'' and ''DAB'' have a common point<ref>{{Cite web |url=https://imomath.com/othercomp/Bra/BraMO04.pdf |title=IMOmath, "26-th Brazilian Mathematical Olympiad 2004" |access-date=2020-01-06 |archive-url=https://web.archive.org/web/20161018164829/http://imomath.com/othercomp/Bra/BraMO04.pdf |archive-date=2016-10-18 |url-status=live }}</ref>

==Basic properties==
Every rhombus has two [[diagonal]]s connecting pairs of opposite vertices, and two pairs of parallel sides. Using [[congruence (geometry)|congruent]] [[triangle]]s, one can [[mathematical proof|prove]] that the rhombus is [[symmetry|symmetric]] across each of these diagonals. It follows that any rhombus has the following properties:
* Opposite [[angle]]s of a rhombus have equal measure.
* The two diagonals of a rhombus are [[perpendicular]]; that is, a rhombus is an [[orthodiagonal quadrilateral]].
* Its diagonals bisect opposite angles.

The first property implies that every rhombus is a [[parallelogram]]. A rhombus therefore has all of the [[Parallelogram#Properties|properties of a parallelogram]]: for example, opposite sides are parallel; adjacent angles are [[supplementary angles|supplementary]]; the two diagonals [[bisection|bisect]] one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the [[parallelogram law]]). Thus denoting the common side as ''a'' and the diagonals as ''p'' and ''q'', in every rhombus
:<math>\displaystyle 4a^2=p^2+q^2.</math>

Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a [[kite (geometry)|kite]]. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.

A rhombus is a [[tangential quadrilateral]].<ref name=Mathworld>{{mathworld |urlname=Rhombus |title=Rhombus}}</ref> That is, it has an [[inscribed figure|inscribed circle]] that is tangent to all four sides.

[[File:Rhombus1.svg|thumb|280px|A rhombus. Each angle marked with a black dot is a right angle. The height ''h'' is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths ''p'' and ''q'' are the red dotted line segments.]]

==Diagonals==
The length of the diagonals ''p = AC'' and ''q = BD'' can be expressed in terms of the rhombus side ''a'' and one vertex angle ''α'' as
:<math>p=a\sqrt{2+2\cos{\alpha}}</math>
and
:<math>q=a\sqrt{2-2\cos{\alpha}}.</math>
These formulas are a direct consequence of the [[law of cosines]].

==Inradius==
The inradius (the radius of a circle [[inscribed]] in the rhombus), denoted by {{math|''r''}}, can be expressed in terms of the diagonals {{math|''p''}} and {{math|''q''}} as<ref name=Mathworld/>
:<math>r = \frac{p \cdot q}{2\sqrt{p^2+q^2}},</math>
or in terms of the side length {{math|''a''}} and any vertex angle {{math|''α''}} or {{math|''β''}} as
:<math>r = \frac{a\sin\alpha}{2} = \frac{a\sin\beta}{2}.</math>

==Area==
As for all [[parallelogram]]s, the [[area]] ''K'' of a rhombus is the product of its [[Base (geometry)|base]] and its [[height]] (''h''). The base is simply any side length ''a'':
:<math>K = a \cdot h .</math>

The area can also be expressed as the [[Edge (geometry)|base]] [[Square (algebra)|squared]] times the sine of any angle:
:<math>K = a^2 \cdot \sin \alpha = a^2 \cdot \sin \beta ,</math>

or in terms of the height and a [[Vertex (geometry)|vertex]] [[angle]]:
:<math>K=\frac{h^2}{\sin\alpha} ,</math>

or as half the product of the [[diagonal]]s ''p'', ''q'':
:<math>K = \frac{p \cdot q}{2} ,</math>

or as the [[semiperimeter]] times the [[radius]] of the [[circle]] [[Inscribed figure|inscribed]] in the rhombus (inradius):
:<math>K = 2a \cdot r .</math>

Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a [[bivector]], so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is the [[determinant]] of the two vectors' Cartesian coordinates: ''K'' = ''x''<sub>1</sub>''y''<sub>2</sub> – ''x''<sub>2</sub>''y''<sub>1</sub>.<ref>[https://www.youtube.com/watch?v=6XghF70fqkY WildLinAlg episode 4] {{Webarchive|url=https://web.archive.org/web/20170205162901/https://www.youtube.com/watch?v=6XghF70fqkY |date=2017-02-05 }}, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube</ref>

==Dual properties==
The [[dual polygon]] of a rhombus is a [[rectangle]]:<ref>de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", ''[[Mathematical Gazette]]'' 95, March 2011, 102-107.</ref>
*A rhombus has all sides equal, while a rectangle has all angles equal.
*A rhombus has opposite angles equal, while a rectangle has opposite sides equal.
*A rhombus has an inscribed circle, while a rectangle has a [[circumcircle]].
*A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides.
*The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.
*The figure formed by joining the midpoints of the sides of a rhombus is a [[rectangle]], and vice versa.

==Cartesian equation==
The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (''x, y'') satisfying

:<math>\left|\frac{x}{a}\right|\! + \left|\frac{y}{b}\right|\! = 1.</math>

The vertices are at <math>(\pm a, 0)</math> and <math>(0, \pm b).</math> This is a special case of the [[superellipse]], with exponent 1.

==Other properties==
*One of the five 2D [[lattice (group)|lattice]] types is the rhombic lattice, also called [[centered rectangular lattice]].
* Rhombi can tile the 2D plane edge-to-edge and periodically in three different ways, including, for the 60° rhombus, the [[rhombille tiling]].
**
{| class=wikitable
!colspan=2|As topological [[square tiling]]s
!As 30-60 degree [[rhombille]] tiling
|-
|[[File:Lattice of rhombuses.svg|240px]]
|[[File:Isohedral tiling p4-51c.svg|242px]]
|[[File:Rhombic star tiling.svg|154px]]
|}

* Three-dimensional analogues of a rhombus include the [[bipyramid]] and the [[bicone]] as a [[surface of revolution]].

===As the faces of a polyhedron===
Convex polyhedra with rhombi include the infinite set of rhombic [[zonohedron]]s, which can be seen as projective envelopes of [[hypercube]]s.

* A [[rhombohedron]] (also called a rhombic hexahedron) is a three-dimensional figure like a [[cuboid]] (also called a rectangular parallelepiped), except that its 3 pairs of parallel faces are up to 3 types of rhombi instead of rectangles.
* The [[rhombic dodecahedron]] is a [[convex polyhedron]] with 12 [[congruence (geometry)|congruent]] rhombi as its [[face (geometry)|faces]].
* The [[rhombic triacontahedron]] is a [[convex polyhedron]] with 30 [[golden rhombus|golden rhombi]] (rhombi whose diagonals are in the [[golden ratio]]) as its faces.
* The [[great rhombic triacontahedron]] is a nonconvex [[Isohedral figure|isohedral]], [[isotoxal]] [[polyhedron]] with 30 intersecting rhombic faces.
* The [[rhombic hexecontahedron]] is a [[stellation]] of the rhombic triacontahedron. It is nonconvex with 60 [[golden rhombus|golden rhombic]] faces with [[icosahedral symmetry]].
* The [[rhombic enneacontahedron]] is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones.
* The [[rhombic icosahedron]] is a polyhedron composed of 20 rhombic faces, of which three, four, or five meet at each vertex. It has 10 faces on the polar axis with 10 faces following the equator.

{| class=wikitable width=640
|+ Example polyhedra with all rhombic faces
! colspan="2"|Isohedral
! colspan="2" |Isohedral golden rhombi
!2-isohedral
!3-isohedral
|- align="center"
|[[File:TrigonalTrapezohedron.svg|40px]]
|[[File:Rhombicdodecahedron.jpg|100px]]
|[[File:Rhombictriacontahedron.svg|100px]]
|[[File:Rhombic icosahedron.png|100px]]
|[[File:Rhombic enneacontahedron.png|100px]]
|[[File:Rhombohedron.svg|100px]]
|- align="center"
![[Trigonal trapezohedron]]
![[Rhombic dodecahedron]]
![[Rhombic triacontahedron]]
![[Rhombic icosahedron]]
![[Rhombic enneacontahedron]]
![[Rhombohedron]]
|}

==See also==
*[[Merkel-Raute]]
*[[Rhombus of Michaelis]], in human anatomy
*[[Rhomboid]], either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle
*[[Rhombic antenna]]
*[[Rhombic Chess]]
*[[Flag of the Department of North Santander]] of Colombia, containing four stars in the shape of a rhombus
*[[Superellipse]] (includes a rhombus with rounded corners)

==References==
{{Reflist}}

==External links==
{{wiktionary}}
{{Commons category}}
*[http://www.elsy.at/kurse/index.php?kurs=Parallelogram+and+Rhombus&status=public Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)]
*[http://www.mathopenref.com/rhombus.html Rhombus definition, Math Open Reference] with interactive applet.
*[http://www.mathopenref.com/rhombusarea.html Rhombus area, Math Open Reference] - shows three different ways to compute the area of a rhombus, with interactive applet

{{Polygons}}
{{Authority control}}

[[Category:Types of quadrilaterals]]
[[Category:Elementary shapes]]