# Definition:Quadrilateral/Trapezium

## Definition

A **trapezium** is a quadrilateral with no parallel sides.

## Also defined as

Outside the US (one of a few countries that use this definition), a **trapezium** is a quadrilateral which has one pair of sides parallel, that is, what the US defines as a trapezoid.

## Also known as

Outside the US (one of a few countries that use this definition), this figure is known as a **trapezoid**.

In order to reduce confusion, when such a quadrilateral is intended, it is probably better to use the term **irregular quadrilateral** instead.

Euclid, in his definitions, did not distinguish between **trapezia** and **trapezoids**.

## Euclid's Definitions

In the words of Euclid:

*Of quadrilateral figures, a***square**is that which is both equilateral and right-angled; an**oblong**that which is right-angled but not equilateral; a**rhombus**that which is equilateral but not right-angled; and a**rhomboid**that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called**trapezia**.

(*The Elements*: Book $\text{I}$: Definition $22$)

## Linguistic Note

The plural of **trapezium** is **trapezia**.

The word comes from Latin, in which language it is a neuter noun of the second declension, hence its plural form.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**trapezium**:**2.**(*mainly North American usage. UK term:***trapezoid**.) - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**trapezoid**