Rhombus
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[[Image:Rhombus.svg|frame|right|Two rhombi.]]
In [[geometry]], a '''rhombus''' (from [[Ancient Greek]] '''''ῥόμβος''''' - rrhombos, “rhombus, spinning top”), (plural '''rhombi''' or '''rhombuses''') or '''rhomb''' (plural '''rhombs''') is an [[equilateral polygon|equilateral]] [[quadrilateral]]. In other words, it is a four-sided polygon in which every side has the same length.
The rhombus is often casually called a diamond, after the [[Diamonds (suit)|diamonds]] suit in playing cards, or a [[lozenge]], because those shapes are rhombi (though not all rhombi are actually diamonds or lozenges).
===Supersets===
In any rhombus, opposite sides are [[parallel (geometry)|parallel]]. Thus, the rhombus is a special case of the [[parallelogram]]. One analogy holds that the rhombus is to the parallelogram as the square is to the rectangle. In its turn, the square is a special case of the rhombus, being most readily defined as a rhombus with one right angle.
A rhombus is also a special case of a [[kite (geometry)|kite]] (a quadrilateral with two distinct pairs of adjacent sides of equal lengths). The opposite sides of a kite are not parallel unless the kite is also a rhombus.
===Area===
The [[area]] of any rhombus is the product of the lengths of its [[diagonal]]s divided by two:<br>
<math>Area=({D_1 \times D_2}) /2</math><br>
Because the rhombus is a [[parallelogram]], the area also equals the length of a side (B) multiplied by the perpendicular distance between two opposite sides(H)
<math>Area=B \times H</math><br>
The area also equals the square of the side multiplied by the sine of any of the exterior angles:<br>
<math>Area=a^2 \sin\theta</math><br>
where a is the length of the side and <math>\theta</math> is the angle between two sides.
===A proof that the diagonals are perpendicular===
One of the five 2D [[lattice (group)|lattice]] types is the rhombic lattice, also called centered rectangular lattice.
If A, B, C and D were the [[vertex (geometry)|vertices]] of the rhombus, named in agreement with the figure (higher on this page). Using <math>\overrightarrow{AB}</math> to represent the [[Vector (spatial)|vector]] from A to B, one notices that
<br><math>\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}</math>
<br><math>\overrightarrow{BD} = \overrightarrow{BC}+ \overrightarrow{CD}= \overrightarrow{BC}- \overrightarrow{AB}</math>.<br>
The last equality comes from the parallelism of CD and AB.
Taking the [[inner product]],
:<math>
<\overrightarrow{AC}, \overrightarrow{BD}> = <\overrightarrow{AB} + \overrightarrow{BC}, \overrightarrow{BC} - \overrightarrow{AB}></math>
::<math>= <\overrightarrow{AB}, \overrightarrow{BC}> - <\overrightarrow{AB}, \overrightarrow{AB}> + <\overrightarrow{BC}, \overrightarrow{BC}> - <\overrightarrow{BC}, \overrightarrow{AB}></math>
::<math> = 0</math>
since the norms of AB and BC are equal and since the inner product is [[bilinear]] and symmetric. The inner product of the diagonals is zero [[if and only if]] they are [[perpendicular]].
===Tilings===
{| class=wikitable
| [[Image:kah_3_6_romb.png|200px]] <br> Rhombic [[tessellation|tiling]]
|}
== Origin ==
The word ''rhombus'' is from the [[Greek language|Greek]] word for something that spins. [[Euclid]] used ''ρόμβος'' (''rhombos''), from the verb ''ρέμβω'' (''rhembo''), meaning "to turn round and round".<ref>[http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2392686 '''Rhombos''', "A Greek-English Lexicon", Liddel and Scott, at Perseus]</ref><ref>[http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2392296 '''Rhembo''', "A Greek-English Lexicon", Liddel and Scott, at Perseus]</ref> [[Archimedes]] used the term "solid rhombus" for two right circular [[cone (geometry)|cone]]s sharing a common base.<ref>http://www.pballew.net/rhomb.html MathWords web page for Rhombus</ref>
==References==
{{reflist}}
== External links ==
{{wiktionary}}
{{commonscat}}
* [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:Greek loanwords]]
[[ar:دالتون(رياضيات)]]
[[ast:Rombu]]
[[az:Romb]]
[[be:Ромб]]
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[[bs:Romb]]
[[bg:Ромб]]
[[ca:Rombe]]
[[cs:Kosočtverec]]
[[da:Rombe]]
[[de:Raute]]
[[et:Romb]]
[[el:Ρόμβος]]
[[es:Rombo]]
[[eo:Rombo]]
[[fr:Losange]]
[[gl:Rombo]]
[[ko:마름모]]
[[hr:Romb]]
[[id:Belah ketupat]]
[[it:Rombo (geometria)]]
[[he:מעוין]]
[[ka:რომბი]]
[[ht:Lozanj]]
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[[lt:Rombas]]
[[li:Roet]]
[[hu:Rombusz]]
[[mn:Ромб]]
[[nl:Ruit (meetkunde)]]
[[ja:菱形]]
[[no:Rombe]]
[[km:ចតុកោណស្មើ]]
[[nds:Ruut]]
[[pl:Romb]]
[[pt:Losango]]
[[ro:Romb]]
[[qu:Puytu]]
[[ru:Ромб]]
[[scn:Rummu]]
[[simple:Rhombus]]
[[sl:Romb]]
[[sr:Ромб]]
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[[sv:Romb]]
[[vi:Hình thoi]]
[[uk:Ромб]]
[[vls:Rute]]
[[zh:菱形]]