Trapezoidal

A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().

A trapezoid is necessarily a convex quadrilateral in Euclidean geometry. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are called the ''legs'' (or the ''lateral sides'') if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A ''scalene trapezoid'' is a trapezoid with no sides of equal measure, in contrast with the special cases below.

Etymology and ''trapezium'' versus ''trapezoid''

Ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (''trapezia'' literally "a table", itself from τετράς (''tetrás''), "four" + πέζα (''péza''), "a foot; end, border, edge").

Two types of ''trapezia'' were introduced by Proclus (412 to 485 AD) in his commentary on the first book of Euclid's Elements:
* one pair of parallel sides – a ''trapezium'' (τραπέζιον), divided into isosceles (equal legs) and scalene (unequal) trapezia
* no parallel sides – ''trapezoid'' (τραπεζοειδή, ''trapezoeidé'', literally trapezium-like ( εἶδος means "resembles"), in the same way as cuboid means cube-like and rhomboid means rhombus-like)

All European languages follow Proclus's structure as did English until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation a transposition of the terms. This mistake was corrected in British English in about 1875, but was retained in American English into the modern day.

The following is a table comparing usages, with the most specific definitions at the top to the most general at the bottom.

Inclusive vs exclusive definition

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. Others define a trapezoid as a quadrilateral with ''at least'' one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.

Under the inclusive definition, all parallelograms (including rhombuses, squares and non-square rectangles) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.

Special cases

A right trapezoid (also called ''right-angled trapezoid'') has two adjacent right angles. Right trapezoids are used in the trapezoidal rule for estimating areas under a curve.

An acute trapezoid has two adjacent acute angles on its longer ''base'' edge, while an obtuse trapezoid has one acute and one obtuse angle on each ''base''.

An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry. This is possible for acute trapezoids or right trapezoids (rectangles).

A parallelogram is a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse trapezoids or right trapezoids (rectangles).

A tangential trapezoid is a trapezoid that has an incircle.

A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the hyperbolic plane has 3 right angles.

Condition of existence

Four lengths ''a'', ''c'', ''b'', ''d'' can constitute the consecutive sides of a non-parallelogram trapezoid with ''a'' and ''b'' parallel only when
:$\displaystyle , d-c, < , b-a, < d+c.$

The quadrilateral is a parallelogram when $d-c = b-a = 0$, but it is an ex-tangential quadrilateral (which is not a trapezoid) when $, d-c, = , b-a, \neq 0$.

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 to 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 have equal areas.Martin Josefsson
"Characterizations of trapezoids"
Forum Geometricorum, 13 (2013) 23-35.
*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.
*The areas ''S'' and ''T'' of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation

::$\sqrt=\sqrt+\sqrt,$

:where ''K'' is the area of the quadrilateral.

*The midpoints of two opposite sides and the intersection of the diagonals are collinear.
*The angles in the quadrilateral ''ABCD'' satisfy $\sin A\sin C=\sin B\sin D.$
*The cosines of two adjacent angles sum to 0, as do the cosines of the other two angles.
*The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.
*One bimedian divides the quadrilateral into two quadrilaterals of equal areas.