Why do trapezium and trapezoid have inverted meanings in the US?

Everywhere in the world:

Trapezium: convex quadrilateral with a pair of parallel sides.

Trapezoid: convex quadrilateral.

US: transpose the above meanings.

As we can see the US interpretation makes $0$ sense. What is the origin of this foul joke?

Solution 1:

According to the "Earliest Known Uses of Some of the Words of Mathematics" site,

TRAPEZIUM and TRAPEZOID. [...] Both trapezium and trapezoid were used by Proclus (c. 410-485). From the time of Proclus until the end of the 18th century, a trapezium was a quadrilateral with two sides parallel and a trapezoid was a quadrilateral with no sides parallel. However, in 1795 a Mathematical and Philosophical Dictionary by Charles Hutton (1737-1823) appeared with the definitions of the two terms reversed:

Trapezium ... a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid.

No previous use the words with Hutton’s definitions is known. Nevertheless, the newer meanings of the two words now prevail in U.S. but not necessarily in Great Britain (OED2).