# Definition:Quadrilateral/Trapezium

## Definition

### Definition $1$

A **trapezium** is a quadrilateral which has exactly one pair of sides that are parallel.

### Definition $2$

A **trapezium** is a quadrilateral which has $2$ parallel sides whose lengths are unequal.

Thus, by this definition, a parallelogram is *not* a **trapezium**.

### Base

The **bases** of a **trapezium** are its $2$ parallel sides.

In the above diagram, the **bases** of the given **trapezia** are $AB$ and $DC$, $EF$ and $HG$, and $IJ$ and $KL$.

### Leg

The **legs** of a **trapezium** are its $2$ sides adjacent to the bases.

In the above diagram, the **legs** of the given **trapezia** are $AD$ and $BC$, $EH$ and $FG$, and $IK$ and $JL$.

### Height

The **height** of a **trapezium** is defined as the length of a line perpendicular to the bases.

In the above diagram, the **heights** of the given **trapezia** are indicated by the letter $h$.

## Usage Differences

The North American definitions of **trapezium** and **trapezoid** differ from most of the rest of the world as follows:

- a
**trapezoid**has one pair of sides that are parallel - a
**trapezium**does*not*have a pair of parallel sides.

This is the opposite way round from the definitions as used in most of the rest of the world, as used by $\mathsf{Pr} \infty \mathsf{fWiki}$.

In order to reduce confusion, when a **trapezoid** is intended, it may be better to use the term **irregular quadrilateral** instead of **trapezoid**.

It is worth noting that Euclid, in his definitions, did not distinguish between **trapezia** and **trapezoids**, and lumped them together as **trapezia**:

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 $22$)

## Also see

- Results about
**trapezia**can be found**here**.

## 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**:**1.**(*mainly UK usage. North American term:***trapezoid**.)