Trapezoid

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For other uses, see Trapezoid (disambiguation).
Trapezoid (AmE)
Trapezium (BrE)
Trapezoid.svg
Trapezoid
Type quadrilateral
Edges and vertices 4
Area \tfrac{a + b}{2} h
Properties convex

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid (pronounced: /ˈtɹæpəzɔɪd/)[1] in 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 special cases below.

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.[2] Others[3] define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition[4]), 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 taxonomy of quadrilaterals.

Etymology[edit]

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.[5]

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]

Special cases[edit]

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),[6] two adjacent angles are right angles.[3] A tangential trapezoid is a trapezoid that has an incircle.

A trapezoid with two pairs of parallel sides is a parallelogram. Under the inclusive definition, all parallelograms (including rhombuses, rectangles and squares) are trapezoids.

Characterizations[edit]

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:

  • The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
  • The diagonals cut each other in mutually the same ratio (this ratio is the same as that between the lengths of the parallel sides).
  • The diagonals cut the quadrilateral into four triangles of which one opposite pair are similar.
  • The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas.[7]:Prop.5
  • 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.[7]:Thm.6
  • The areas S and T of some two opposite triangles of the four triangles formed by the diagonals verify the equation
\sqrt{K}=\sqrt{S}+\sqrt{T},
where K is the area of the quadrilateral.[7]:Thm.8
  • The midpoints of two opposite sides and the intersection of the diagonals are collinear.[7]:Thm.15

Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel:

  • The consecutive sides a, c, b, d and the diagonals p, q verify the equation[7]:Cor.11
p^2+q^2=c^2+d^2+2ab.
  • The distance v between the midpoints of the diagonals verifies the equation[7]:Thm.12
v=\frac{|a-b|}{2}.

Midsegment and height[edit]

The midsegment (also called the median or midline) of a trapezoid is the segment that joins the midpoints 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,[3]

m = \frac{a + b}{2}.

The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides the trapezoid into equal areas).

The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths (ab), the height of a trapezoid h can be determined by the length of its four sides using the formula[3]

h= \frac{\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}

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[8]

\displaystyle h^2>0.

Area[edit]

The area K of a trapezoid is given by[3]

K = \frac{a + b}{2} \cdot h = mh

where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and m is the arithmetic mean of the lengths of the two parallel 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.

From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as[3]

K = \frac{a+b}{4|b-a|}\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.

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.

Another equivalent formula for the area, which more closely resembles Heron's formula, is[3]

K = \frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)},

where s = \tfrac{1}{2}(a + b + c + d) 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 (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).

From Bretschneider's formula, it follows that

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}.

The line that joins the midpoints of the parallel sides, bisects the area.

Diagonals[edit]

Trapezium.svg

The lengths of the diagonals are[3]

p= \sqrt{\frac{ab^2-a^2b-ac^2+bd^2}{b-a}},
q= \sqrt{\frac{ab^2-a^2b-ad^2+bc^2}{b-a}}

where a and b are the bases, c and d are the other two sides, and a < b.

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 \triangle AOD is equal to that of \triangle BOC, and the product of the areas of \triangle AOD and \triangle BOC is equal to that of \triangle AOB and \triangle COD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.[3]

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:[9]

\frac{1}{FG}=\frac{1}{2} \left( \frac{1}{AB}+ \frac{1}{DC} \right).

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.[10]

Other properties[edit]

The center of area (center of mass for a uniform lamina) lies along the line joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by[11]

x = \frac{h}{3} \left( \frac{2a+b}{a+b}\right).

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[10]

PQ=\frac{|AD+BC-AB-CD|}{2}.

More on terminology[edit]

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.[12][13] 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".)[14]

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.[14]

Architecture[edit]

The Temple of Dendur in the Metropolitan Museum of Art, New York

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 trapezoids. This was the standard style for the doors and windows of the Incas.[15]

See also[edit]

References[edit]

  1. ^ "trapezoid - Wiktionary". 
  2. ^ "American School definition from "math.com"". Retrieved 2008-04-14. 
  3. ^ a b c d e f g h i Weisstein, Eric W., "Trapezoid", MathWorld.
  4. ^ Trapezoids, [1], accessed 2012-02-24.
  5. ^ Oxford English Dictionary entry at trapezoid.
  6. ^ [2], picture
  7. ^ a b c d e f Martin Josefsson, "Characterizations of trapezoids", Forum Geometricorum, 13 (2013) 23-35.
  8. ^ Quadrilateral Formulas, The Math Forum, Drexel University, 2012, [3].
  9. ^ GoGeometry, [4], Accessed 2012-07-08.
  10. ^ a b Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 55.
  11. ^ efunda, General Trapezoid, [5], Accessed 2012-07-09.
  12. ^ Chambers 21st Century Dictionary Trapezoid
  13. ^ "1913 American definition of trapezium". Merriam-Webster Online Dictionary. Retrieved 2007-12-10. 
  14. ^ a b Oxford English Dictionary entries for trapezoid and trapezium.
  15. ^ http://gogeometry.com/MachuPicchu.htm

External links[edit]