A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases.

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.

This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid.

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

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[9] literally 'table', itself from τετράς (tetrás) 'four' + πέζα (péza) 'foot; end, border, edge').

[10] Two types of trapezia were introduced by Proclus (AD 412 to 485) in his commentary on the first book of Euclid's Elements:[8][11] All other European languages follow Proclus's structure[11][12] 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.

[8] The following table compares usages, with the most specific definitions at the top to the most general at the bottom.

A parallelogram is (under the inclusive definition) a trapezoid with two pairs of parallel sides.

Four lengths a, c, b, d can constitute the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when[16] The quadrilateral is a parallelogram when

35 Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel: The midsegment of a trapezoid is the segment that joins the midpoints of the legs.

The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides a, c, b, d: where a and b are parallel and b > a.

From Bretschneider's formula, it follows that The bimedian connecting the parallel sides bisects the area.

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

The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.

[15] Let the trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and DC.

[20] The center of area (center of mass for a uniform lamina) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by[21] The center of area divides this segment in the ratio (when taken from the short to the long side)[22]: p. 862 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[20] In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style.

If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids.

[23] The crossed ladders problem is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection.

Typical designs will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent.