In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides.
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.
A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.
Etymology and trapezium versus trapezoid
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').
Two types of trapezia were introduced by Proclus (AD 412 to 485) 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 was reversed in British English in about 1875, but it has been retained in American English to the present.
The following table compares usages, with the most specific definitions at the top to the most general at the bottom.
Inclusive versus 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. 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.
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.
An obtuse trapezoid on the other hand 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 (as rectangles).
A parallelogram is (under the inclusive definition) 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
The quadrilateral is a parallelogram when , but it is an ex-tangential quadrilateral (which is not a trapezoid) when .