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| {{For||Trapezoid (disambiguation)}}
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| {{Infobox Polygon
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| | name = Trapezoid (AmE)<br>Trapezium (BrE)
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| | image = Trapezoid.svg
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| | caption = Trapezoid
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| | type = [[quadrilateral]]
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| | edges = 4
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| | symmetry =
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| | area = <math>\tfrac{a + b}{2} h</math>
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| | properties = [[convex polygon|convex]]}}
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| In [[Euclidean geometry]], a [[convex set|convex]] [[quadrilateral]] with at least one pair of parallel sides is referred to as a '''trapezoid''' in [[American English|American]] and [[Canadian English]] but as a '''trapezium''' in English outside North America. The parallel sides are called the ''bases'' of the trapezoid and the other two sides are called the ''legs'' or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A ''scalene trapezoid'' is a trapezoid with no sides of equal measure, in contrast to the [[Trapezoid#Special cases|special cases]] below.
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| There is some disagreement whether [[parallelogram]]s, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having ''only'' one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.<ref>{{cite web |url=http://www.math.com/school/glossary/defs/trapezoid.html |title=American School definition from "math.com" |accessdate=2008-04-14}}</ref> Others<ref name=Mathworld>{{MathWorld |title=Trapezoid |urlname=Trapezoid}}</ref> define a trapezoid as a quadrilateral with ''at least'' one pair of parallel sides (the inclusive definition<ref>Trapezoids, [http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html], accessed 2012-02-24.</ref>), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the [[trapezoidal approximation]] to a [[definite integral]] ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the [[Quadrilateral#Taxonomy|taxonomy of quadrilaterals]].
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| ==Etymology==
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| The term ''trapezium'' has been in use in English since 1570, from Late Latin ''trapezium'', from Greek τραπέζιον (''trapézion''), literally "a little table", a diminutive of τράπεζα (''trápeza''), "a table", itself from τετράς (''tetrás''), "four" + πέζα (''péza''), "a foot, an edge". The first recorded use of the Greek word translated ''trapezoid'' (τραπέζοειδη, ''trapézoeide'', "table-like") was by Marinus [[Proclus]] (412 to 485 AD) in his Commentary on the first book of [[Euclid's Elements]].<ref>Oxford English Dictionary entry at ''trapezoid''.</ref> | |
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| This article uses the term ''trapezoid'' in the sense that is current in the United States and Canada. In all other languages using a word derived from the Greek for this figure, the form closest to ''trapezium'' (e.g. French ''trapèze'', Italian ''trapezio'', Spanish ''trapecio'', German ''Trapez'', Russian ''трапеция'') is used.{{citation needed|date=January 2013}}
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| ==Special cases==
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| In an [[isosceles trapezoid]], the legs (''AD'' and ''BC'' in the figure above) have the same length, and the base angles have the same measure. In a '''right trapezoid''' (also called right-angled trapezoid), two adjacent angles are [[right angles]].<ref name=Mathworld/> A [[tangential trapezoid]] is a trapezoid that has an [[Tangential quadrilateral|incircle]].
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| A trapezoid with two pairs of parallel sides is a [[parallelogram]]. Under the inclusive definition, all parallelograms (including [[rhombus]]es, [[rectangle]]s and [[Square (geometry)|square]]s) are trapezoids.
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| ==Characterizations==
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| Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
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| *It has two adjacent [[angle]]s that are [[supplementary angles|supplementary]], that is, they add up 180 [[degree (angle)|degree]]s.
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| *The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
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| *The [[diagonal]]s cut each other in mutually the same [[ratio]] (this ratio is the same as that between the lengths of the parallel sides).
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| *The diagonals cut the quadrilateral into four triangles of which one opposite pair are [[similarity (geometry)|similar]].
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| *The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas.<ref name=Josefsson>Martin Josefsson, [http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf "Characterizations of trapezoids"], Forum Geometricorum, 13 (2013) 23-35.</ref>{{rp|Prop.5}}
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| *The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.<ref name=Josefsson/>{{rp|Thm.6}}
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| *The areas ''S'' and ''T'' of some two opposite triangles of the four triangles formed by the diagonals verify the equation
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| ::<math>\sqrt{K}=\sqrt{S}+\sqrt{T},</math>
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| :where ''K'' is the area of the quadrilateral.<ref name="Josefsson" />{{rp|Thm.8}}
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| *The midpoints of two opposite sides and the intersection of the diagonals are [[collinear]].<ref name="Josefsson" />{{rp|Thm.15}}
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| Additionally, the following properties are equivalent, and each implies that opposite sides ''a'' and ''b'' are parallel:
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| * The consecutive sides ''a'', ''c'', ''b'', ''d'' and the diagonals ''p'', ''q'' verify the equation <ref name="Josefsson" />{{rp|Cor.11}}
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| ::<math>p^2+q^2=c^2+d^2+2ab.</math>
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| *The distance ''v'' between the midpoints of the diagonals verifies the equation<ref name="Josefsson" />{{rp|Thm.12}}
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| ::<math>v=\frac{|a-b|}{2}.</math>
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| ==Midsegment and height==
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| The ''midsegment'' (also called the median or midline) of a trapezoid is the segment that joins the [[midpoint]]s of the legs. It is parallel to the bases. Its length ''m'' is equal to the average of the lengths of the bases ''a'' and ''b'' of the trapezoid,<ref name=Mathworld/>
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| :<math>m = \frac{a + b}{2}.</math>
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| The midsegment of a trapezoid is one of the two [[Quadrilateral#Special line segments|bimedian]]s (the other bimedian divides the trapezoid into equal areas).
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| The ''height'' (or altitude) is the [[perpendicular]] distance between the bases. In the case that the two bases have different lengths (''a'' ≠ ''b''), the height of a trapezoid ''h'' can be determined by the length of its four sides using the formula<ref name=Mathworld/>
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| :<math>h= \frac{\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}</math>
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| where ''c'' and ''d'' are the lengths of the legs. This formula also gives a way of determining when a trapezoid of consecutive sides ''a'', ''c'', ''b'', and ''d'' exists. There is such a trapezoid with bases ''a'' and ''b'' if and only if<ref>Quadrilateral Formulas, ''The Math Forum'', Drexel University, 2012, [http://mathforum.org/dr.math/faq/formulas/faq.quad.html].</ref>
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| :<math>\displaystyle h^2>0.</math>
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| ==Area==
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| The area ''K'' of a trapezoid is given by<ref name=Mathworld/>
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| :<math>K = \frac{a + b}{2} \cdot h</math>
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| where ''a'' and ''b'' are the lengths of the parallel sides, and ''h'' is the height (the perpendicular distance between these sides.) In 499 AD [[Aryabhata]], a great [[mathematician]]-[[astronomer]] from the classical age of [[Indian mathematics]] and [[Indian astronomy]], used this method in the ''[[Aryabhatiya]]'' (section 2.8). This yields as a special case the well-known formula for the area of a [[triangle]], by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.
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| Therefore the area of a trapezoid is equal to the length of this midsegment multiplied by the height<ref name=Mathworld/>
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| :<math>K = mh\,</math> | |
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| From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as<ref name=Mathworld/>
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| :<math>K = \frac{a+b}{4|b-a|}\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.</math>
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| When one of the parallel sides has shrunk to a point (say ''a'' = 0), this formula reduces to [[Heron's formula]] for the area of a triangle.
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| Another equivalent formula for the area, which more closely resembles Heron's formula, is<ref name=Mathworld/>
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| :<math>K = \frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)},</math>
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| where <math>s = \tfrac{1}{2}(a + b + c + d)</math> is the [[semiperimeter]] of the trapezoid. (This formula is similar to [[Brahmagupta's formula]], but it differs from it, in that a trapezoid might not be [[cyclic quadrilateral|cyclic]] (inscribed in a circle). The formula is also a special case of [[Bretschneider's formula]] for a general [[quadrilateral]]).
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| From Bretschneider's formula, it follows that
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| :<math>K= \sqrt{\frac{(ab^2-a^2 b-ad^2+bc^2)(ab^2-a^2 b-ac^2+bd^2)}{(2(b-a))^2} - \left(\frac{b^2+d^2-a^2-c^2}{4}\right)^2}.</math>
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| The line that joins the midpoints of the parallel sides, bisects the area.
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| ==Diagonals==
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| [[File:Trapezium.svg|200px|right]]
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| The lengths of the diagonals are<ref name=Mathworld/>
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| :<math>p= \sqrt{\frac{ab^2-a^2b-ac^2+bd^2}{b-a}},</math>
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| :<math>q= \sqrt{\frac{ab^2-a^2b-ad^2+bc^2}{b-a}}</math>
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| where ''a'' and ''b'' are the bases, ''c'' and ''d'' are the other two sides, and ''a'' < ''b''.
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| If the trapezoid is divided into four triangles by its diagonals ''AC'' and ''BD'' (as shown on the right), intersecting at ''O'', then the area of {{trianglenotation|AOD}} is equal to that of {{trianglenotation|BOC}}, and the product of the areas of {{trianglenotation|AOD}} and {{trianglenotation|BOC}} is equal to that of {{trianglenotation|AOB}} and {{trianglenotation|COD}}. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.<ref name=Mathworld/>
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| Let the trapezoid have vertices ''A'', ''B'', ''C'', and ''D'' in sequence and have parallel sides ''AB'' and ''DC''. Let ''E'' be the intersection of the diagonals, and let ''F'' be on side ''DA'' and ''G'' be on side ''BC'' such that ''FEG'' is parallel to ''AB'' and ''CD''. Then ''FG'' is the [[harmonic mean]] of ''AB'' and ''DC'':<ref>''GoGeometry'', [http://www.gogeometry.com/problem/p747-trapezoid-diagonal-parallel-similarity-harmonic-mean-high-school-college.htm], Accessed 2012-07-08.</ref>
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| :<math>\frac{1}{FG}=\frac{1}{2} \left( \frac{1}{AB}+ \frac{1}{DC} \right).</math>
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| The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.<ref name=Byer/>
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| ==Other properties==
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| The center of area (center of mass for a uniform [[planar lamina|lamina]]) lies along the line joining the midpoints of the parallel sides, at a perpendicular distance ''x'' from the longer side ''b'' given by<ref>''efunda'', General Trapezoid, [http://www.efunda.com/math/areas/Trapezoid.cfm], Accessed 2012-07-09.</ref>
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| :<math>x = \frac{h}{3} \left( \frac{2a+b}{a+b}\right).</math>
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| If the angle bisectors to angles ''A'' and ''B'' intersect at ''P'', and the angle bisectors to angles ''C'' and ''D'' intersect at ''Q'', then<ref name=Byer>Owen Byer, Felix Lazebnik and Deirdre Smeltzer, ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, p. 55.</ref>
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| :<math>PQ=\frac{|AD+BC-AB-CD|}{2}.</math>
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| ==More on terminology==
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| The term trapezium is sometimes defined in the USA as a [[quadrilateral]] with no parallel sides, though this shape is more usually called an irregular quadrilateral.<ref>[http://www.chambersharrap.co.uk/chambers/features/chref/chref.py/main?xref=21C44644&title=21st&query=trapezoid ''Chambers 21st Century Dictionary'' Trapezoid]</ref><ref>{{cite web |url=http://www.merriam-webster.com/dictionary/trapezium |title=1913 American definition of trapezium |work=Merriam-Webster Online Dictionary |accessdate=2007-12-10}}</ref> The term trapezoid was once defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage. (The Oxford English Dictionary says "Often called by English writers in the 19th century".)<ref name="oed">''[[Oxford English Dictionary]]'' entries for trapezoid and trapezium.</ref>
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| According to the ''[[Oxford English Dictionary]]'', the sense of a figure with no sides parallel is the meaning for which [[Proclus]] introduced the term "trapezoid". This is retained in the French ''trapézoïde'', German ''Trapezoid'', and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a [[parallelogram]] is the sense of the term in [[Euclid]]. The sense of a trapezium as an irregular quadrilateral having no sides parallel was sometimes used in England from c. 1800 to c. 1875, but is now obsolete. This sense is the one that is sometimes quoted in the US, but in practice quadrilateral is used rather than trapezium.<ref name="oed"/>
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| ==Architecture==
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| [[Image:Temple of Dendur- night.jpg|right|250px|thumb|The [[Temple of Dendur]] in the Metropolitan Museum of Art, New York]]
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| In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering towards the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually [[isosceles trapezoid]]s. This was the standard style for the doors and windows of the [[Inca Empire|Inca]]<nowiki/>s.<ref>http://gogeometry.com/MachuPicchu.htm</ref>
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| ==See also==
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| *[[Polite number]], also known as a trapezoidal number
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| *[[Trapezoidal rule]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.mathopenref.com/trapezoid.html Trapezoid definition] [http://www.mathopenref.com/trapezoidarea.html Area of a trapezoid] [http://www.mathopenref.com/trapezoidmedian.html Median of a trapezoid] With interactive animations
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| *[http://www.elsy.at/kurse/index.php?kurs=Trapezoid+%28North+America%29&status=public Trapezoid (North America)] at elsy.at: Animated course (construction, circumference, area)
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| *[http://numericalmethods.eng.usf.edu/topics/trapezoidal_rule.html] on ''Numerical Methods for Stem Undergraduate''
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| *Autar Kaw and E. Eric Kalu, ''Numerical Methods with Applications'', (2008) [http://www.autarkaw.com/books/numericalmethods/index.html]
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| [[Category:Quadrilaterals]]
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| [[Category:Elementary shapes]]
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