Rhombus 183884 223738322 2008-07-05T15:12:00Z Numbo3-bot 7118326 robot Adding: [[be:Ромб]] {{otheruses}} [[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]] [[be-x-old:Ромб]] [[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]] [[lv:Rombs]] [[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]] [[fi:Neljäkäs]] [[sv:Romb]] [[vi:Hình thoi]] [[uk:Ромб]] [[vls:Rute]] [[zh:菱形]]