### Trapezoidal

In *ABCD*.

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having *exactly* one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.^{[1]} Others^{[2]} define a trapezoid as a quadrilateral with *at least* one pair of parallel sides (the inclusive definition^{[3]}), 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.

## Contents

## Etymology

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.^{[4]}

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.**
**

## Special cases

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.^{[2]} A tangential trapezoid is a trapezoid that has an incircle.

Under the inclusive definition, all parallelograms (including rhombuses, rectangles and squares) are trapezoids.

## Characterizations

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

- It has two adjacent angles that are supplementary, that is, they add up 180 degrees.

- 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.
^{[5]}^{: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.
^{[5]}^{:Thm.6}

- The areas
*S*and*T*of some two opposite triangles of the four triangles formed by the diagonals verify the equation

- $\backslash sqrt\{K\}=\backslash sqrt\{S\}+\backslash sqrt\{T\},$

- where
*K*is the area of the quadrilateral.^{[5]}^{:Thm.8}

- The midpoints of two opposite sides and the intersection of the diagonals are collinear.
^{[5]}^{: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^{[5]}^{:Cor.11}

- $p^2+q^2=c^2+d^2+2ab.$

- The distance
*v*between the midpoints of the diagonals verifies the equation^{[5]}^{:Thm.12}

- $v=\backslash frac\{|a-b|\}\{2\}.$

## Midsegment and height

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,^{[2]}

- $m\; =\; \backslash frac\{a\; +\; b\}\{2\}.$

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 (*a* ≠ *b*), the height of a trapezoid *h* can be determined by the length of its four sides using the formula^{[2]}

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

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^{[6]}

- $\backslash displaystyle\; h^2>0.$

## Area

The area *K* of a trapezoid is given by^{[2]}

- $K\; =\; \backslash frac\{a\; +\; b\}\{2\}\; \backslash cdot\; h$

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).^{[7]} 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.

Therefore the area of a trapezoid is equal to the length of this midsegment multiplied by the height^{[2]}

- $K\; =\; mh\backslash ,$

From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as^{[2]}

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

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^{[2]}

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

where $s\; =\; \backslash tfrac\{1\}\{2\}(a\; +\; b\; +\; c\; +\; d)$ 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

- $K=\; \backslash sqrt\{\backslash frac\{(ab^2-a^2\; b-ad^2+bc^2)(ab^2-a^2\; b-ac^2+bd^2)\}\{(2(b-a))^2\}\; -\; \backslash left(\backslash frac\{b^2+d^2-a^2-c^2\}\{4\}\backslash right)^2\}.$

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

## Diagonals

The lengths of the diagonals are^{[2]}

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

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 $\backslash triangle$ *AOD* is equal to that of $\backslash triangle$ *BOC*, and the product of the areas of $\backslash triangle$ *AOD* and $\backslash triangle$ *BOC* is equal to that of $\backslash triangle$ *AOB* and $\backslash triangle$ *COD*. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.^{[2]}

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*:^{[8]}

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

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.^{[9]}

## Other properties

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^{[10]}

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

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^{[9]}

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

## More on terminology

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.^{[11]}^{[12]} 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".)^{[13]}

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.^{[13]}

## Architecture

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.

## See also

- Polite number, also known as a trapezoidal number
- Trapezoidal rule

## References

## External links

- Median of a trapezoid With interactive animations
- Trapezoid (North America) at elsy.at: Animated course (construction, circumference, area)
- [5] on
*Numerical Methods for Stem Undergraduate* - Autar Kaw and E. Eric Kalu,
*Numerical Methods with Applications*, (2008) [6]