There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; the British definition is a quadrilateral with two sides parallel (e.g., Bronshtein and Semendyayev 1977, p. 174)--which Americans call a trapezoid.
Definitions for trapezoid and trapezium have caused controversy for more than two thousand years.
Euclid (Book 1, Definition 22) stated, "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 opposite sides and angles equal to one another but is neither equilateral nor right angled. And let quadrilaterals other than these be called trapezia."
Proclus (also Heron and Posidonius) divided quadrilaterals into parallelograms and non-parallelograms. For the latter, Proclus assigned trapezium to "two sides parallel," and trapezoid to "no sides parallel." Archimedes also defined a trapezium as having precisely two parallel sides (Heath 1956, pp. 188-190).
According to the Oxford English Dictionary, the confusion of trapezium and trapezoid between the United States and Great Britain dates back to an error in Hutton's Mathematical Dictionary in 1795, the first work of its kind in the United States, which directly reversed the accepted meanings. Hutton assigned trapezium to "no sides parallel" and trapezoid to "two sides parallel" (Simpson and Weiner 1992, p. 2101).
After 1795 in the United States, the Hutton definitions became standard, while in the British empire, the Proclus definitions remained standard. Two hundred years later, the controversy remains. Country by country, region by region, and even teacher by teacher, the definitions of trapezoid and trapezium are commonly swapped.
It is perhaps therefore best to tread extremely carefully into questions of definition for these two simple plane figures. W. E. Greig (pers. comm., Mar. 10, 2007) has proposed that the American trapezoid (i.e., the British trapezium) be dubbed the "trapeziam" (with the -am suffix indicating "American"), but adding yet another term to the word soup seems unlikely to help resolve the confusion.
