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{{Infobox Polygon
| name = rhombus
| image = rhombus.svg
| caption = Two rhombi.
| type = [[quadrilateral]], [[bipyramid]]
| edges = 4
| symmetry = [[dihedral symmetry|Dih<sub>2</sub>]], [2], (*22), order 4
| coxeter = {{CDD|node_f1|2|node_f1}}
| schläfli = { } + { } or 2{ }
| area = <math>\tfrac{pq}{2}</math>
| dual = [[rectangle]]
| properties = [[convex polygon|convex]], [[isotoxal figure|isotoxal]]}}
In [[Euclidean geometry]], a '''rhombus''' (◊), plural '''rhombi''' or '''rhombuses''', is a [[simple polygon|simple]] (non-self-intersecting) [[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 cards, or a '''[[lozenge]]''', though the former sometimes refers specifically to a rhombus with a 60° angle (see [[Polyiamond]]), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a [[parallelogram]], and a rhombus with right angles is a [[Square (geometry)|square]]. <ref>Note: [[Euclid]]'s original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.</ref><ref>{{MathWorld |urlname=Square |title=Square}} inclusive usage</ref>
==Etymology==
The word "rhombus" comes from [[Greek language|Greek]] ῥόμβος (''rhombos''), meaning something that spins,<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dr%28o%2Fmbos ῥόμβος], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> which derives from the verb ρέμβω (''rhembō''), meaning "to turn round and round".<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dr%28e%2Fmbw ρέμβω], 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 two right circular [[cone (geometry)|cone]]s sharing a common base.<ref>[http://www.pballew.net/rhomb The Origin of Rhombus]</ref>

==Characterizations==
A [[simple polygon|simple]] (non self-intersecting) quadrilateral is a rhombus [[if and only if]] it is any one of the following:<ref>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 55-56.</ref><ref>Owen Byer, Felix Lazebnik and Deirdre Smeltzer, ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, p. 53.</ref>
*a quadrilateral with four sides of equal length (by definition)
*a quadrilateral in which the [[diagonal]]s are [[perpendicular]] and [[Bisection|bisect]] each other
*a quadrilateral in which each diagonal bisects two opposite [[Internal and external angle|interior angles]]
*a [[parallelogram]] in which at least two consecutive sides are equal in length
*a parallelogram in which the diagonals are perpendicular
*a parallelogram in which a diagonal bisects an interior angle

==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 of its sides.

==Area==
[[File:Rhombus1.svg|right|280px]]
As for all parallelograms, the [[area]] ''A'' of a rhombus is the product of its base and its height. The base is simply any side length ''b'', and the height ''h'' is the perpendicular distance between any two non-adjacent sides:
:<math>A = b \cdot h .</math>

The area can also be expressed as the base squared times the sine of any angle:
:<math>A = b^2 \cdot \sin \alpha = b^2 \cdot \sin \beta ,</math>

or as half the product of the diagonals ''p'', ''q'':
:<math>A = \frac{p \cdot q}{2} ,</math>

or as the [[semiperimeter]] times the radius of the circle [[Inscribed figure|inscribed]] in the rhombus (inradius):
:<math>A = 2b \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 ( area = x1*y2-x2*y1 ) <ref>[http://www.youtube.com/watch?v=6XghF70fqkY WildLinAlg episode 4], Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube</ref>

==Inradius==
The inradius (the radius of the incircle) can be expressed in terms of the diagonals ''p'' and ''q'' as<ref name=Mathworld/>
:<math>r = \frac{p \cdot q}{2\sqrt{p^2+q^2}}.</math>

==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, in order, the midpoints of the sides of a rhombus is a [[rectangle]] and vice-versa.

==Other properties==
*One of the five 2D [[lattice (group)|lattice]] types is the rhombic lattice, also called [[centered rectangular lattice]]
* Identical rhombi can tile the 2D plane 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:Isohedral_tiling_p4-55.png|240px]]
|[[File:Isohedral_tiling_p4-51c.png|152px]]
|[[File:Rhombic star tiling.png|154px]]
|}

* Three-dimensional analogues of a rhombus include the [[bipyramid]] and the [[bicone]]
* Several [[polyhedra]] have rhombic faces, such as the [[rhombic dodecahedron]] and the [[trapezo-rhombic dodecahedron]]
{| class=wikitable
|+ Some polyhedra with all rhombic faces
!colspan=3|Identical rhombi
!colspan=2|Two types of rhombi
|- align=center
|[[File:Rhombohedron.svg|100px]]
|[[File:Rhombicdodecahedron.jpg|100px]]
|[[File:Rhombictriacontahedron.jpg|100px]]
|[[File:Rhombic icosahedron.png|100px]]
|[[File:Rhombic enneacontahedron.png|100px]]
|- align=center
![[Rhombohedron]]
![[Rhombic dodecahedron]]
![[Rhombic triacontahedron]]
![[Rhombic icosahedron]]
![[Rhombic enneacontahedron]]
|}

==See also==
* [[Rhombus of Michaelis]]

==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.

[[Category:Quadrilaterals]]
[[Category:Elementary shapes]]

Hypericum perforatum - Revision history

71.82.61.9 at 09:40, 11 January 2015

131.93.98.67: /* Adverse effects and drug interactions */ Removed the impersonal you.

en>Rjwilmsi: /* Mechanism of action */Journal cites, added 1 DOI using AWB (9904)