Rectangles have mirror symmetry on mid-edges rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.

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

This is also advocated in the taxonomy of quadrilaterals.

This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. 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. Some sources use the term proper trapezoid to describe trapezoids under the exclusive definition, analogous to uses of the word proper in some other mathematical objects. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Two parallel sides, and no line of symmetry Two parallel sides, and a line of symmetry Opposite sides and angles equal to one another but not equilateral nor right-angled Proclus (Definitions 30-34, quoting Posidonius) The following table compares usages, with the most specific definitions at the top to the most general at the bottom. This was reversed in British English in about 1875, but it has been retained in American English to the present.

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')Īll 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.

one pair of parallel sides – a trapezium (τραπέζιον), divided into isosceles (equal legs) and scalene (unequal) trapezia.

Two types of trapezia were introduced by Proclus (AD 412 to 485) in his commentary on the first book of Euclid's Elements:

The 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 'table', itself from τετράς ( tetrás) 'four' + πέζα ( péza) 'foot end, border, edge'). The metric formulas in this article apply in convex trapezoids.Įtymology and trapezium versus trapezoid Hutton's definitions in 1795 If ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below.Ī trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. 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. The parallel sides are called the bases of the trapezoid. In geometry, a trapezoid ( / ˈ t r æ p ə z ɔɪ d/) in North American English, or trapezium ( / t r ə ˈ p iː z i ə m/) in British English, is a quadrilateral that has at least one pair of parallel sides. Look up trapezoid in Wiktionary, the free dictionary.

Isosceles trapezoid area diagonals