Definition:Quadrilateral/Trapezium

Definition

A trapezium is a quadrilateral which has exactly one pair of sides parallel:

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 trapezium 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 trapezium 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 height of the given trapezia are indicated by the letter $h$.

Also defined as

In the US, a trapezium is a quadrilateral with no parallel sides, that is, what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as a trapezoid.

Also known as

In the US, this figure is known as a trapezoid.

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 $22$)

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.)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trapezium (US: trapezoid)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trapezium (US: trapezoid)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): trapezium (trapezia)