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 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 $h$.
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.)