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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?

Asinomás
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    I blame Noah Webster – Will Jagy Jun 02 '17 at 01:41
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    on the other hand, this is the first time I have heard of a trapezium – Will Jagy Jun 02 '17 at 01:42
  • I thought they were the same thing, but like maths and math we use different words. I have not heard either term applied to and irregular quadrilateral. – Doug M Jun 02 '17 at 01:45
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    People on this site sometimes ask about the sinus. Which is, really, the part of the body that gets congested during a cold. – Will Jagy Jun 02 '17 at 02:10
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    As far as modern American mathematics education is concerned, there's no such thing as a "trapezium." They just go straight from quadrilateral to trapezoid (and then, usually, parallelogram, rectangle, rhombus, and square, totally skipping kites and darts). – Kevin Jun 02 '17 at 02:45
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    It seems unfair to say "everywhere else in the world." That makes it sound as if you were comparing the U.S. to every other country in the world. In fact, there are only seven countries in the world with large numbers of native English-speakers, and of these, two (the U.S. and Canada) use the word "trapezoid" in the way you object to. So it's really 5 to 2. But one of these two countries has the overwhelming majority of the world's English-speakers. – user49640 Jun 02 '17 at 04:27
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    The terms also exist in other languages. – Asinomás Jun 02 '17 at 04:29
  • Each language has its own terminology. If you say "seno", I'm not going to argue that you should say "sino" or "sinus" or anything else to conform to what other languages do. That's a matter for Spanish-speakers to decide. – user49640 Jun 02 '17 at 04:33
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    @user49640 what does that have to do with this? trapecio and trapezoide obviously are the equivalents of trapezium and trapezoid right? and the same happens in many other languages. – Asinomás Jun 02 '17 at 04:44
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    One could logically expect that a trapezoid is something that is like a trapezium but not quite - like a cuboid is like a cube but not aquite, or a humanoid is like a human but not actually one. – user253751 Jun 02 '17 at 06:15
  • @JorgeFernándezHidalgo It would have been more accurate to speak of "equivalents in other European languages" rather than "everywhere in the world," which is an exaggeration. Here is an answer given to this question on English Stack Exchange: https://english.stackexchange.com/questions/65259/trapezium-trapezoid-why-are-the-us-uk-definitions-swapped-around – user49640 Jun 02 '17 at 06:20
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    My American mathematics curriculum definitely covered kites. I think I remember seeing trapezium defined somewhere as a "none of the above" convex quadrilateral, but that may not have been in school. As a child, however, it puzzled me that we used words like "trapezoid" but didn't use "trapeze" (or "trapezium") to describe an isoceles trapezoid(US)/trapezium(UK), which after all is the proper equilibrium shape of a circus trapeze. – David K Jun 03 '17 at 00:37
  • According to the accepted answer and original Proclus' definitions the quadrilateral with no sides parallel (whatever you wish to call it) would not have the 'convex' restriction on it. Such a quadrilateral can be both convex and concave. Take a dart figure as an example of a concave quadrilateral. – Michael Munta Oct 22 '21 at 08:18

1 Answers1

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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).

Community
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Blue
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  • Interesting (+1). This related answer cites a source that claims Hutton's definition was "an error". – dxiv Jun 02 '17 at 02:00
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    https://en.wikipedia.org/wiki/Trapezius_muscle – Will Jagy Jun 02 '17 at 02:07
  • @dxiv: I was going to speculate that Hutton (or perhaps his type-setter) might have made a mistake. After all, his was a reference book, not a textbook or treatise; the terms might well have appeared just once, so that their transposition (trapz-position?) could be overlooked relatively easily. – Blue Jun 02 '17 at 02:25
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    To expand on "not necessarily in Great Britain", my experience in England is that while everyone uses "trapezium" for a quadrilateral with two parallel sides, most people aren't aware of the original meaning of "trapezoid" and just think of it as the American word for trapezium. – Especially Lime Jun 02 '17 at 07:54