Talk:Trapezoid

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[edit] Miscellaneous comments

Perhaps you should have references here. Isn't the rule, No reference. No article? If some don't show up, someone will prod it for deletion! —Preceding unsigned comment added by 69.246.54.228 (talk) 01:53, 19 September 2007 (UTC)

Note that a trapezium (British English) or trapezoid (American English) does not have two parallel sides. What kind of beast is a (U.S.)trapezium ? --FvdP 19:37, 12 Mar 2004 (UTC)

A USA Trapezium is what the rest of the world calls trapezoid (no sides parallel). For whatever reason the Americans swapped the name. — Jor (Talk) 19:20, 1 Apr 2004 (UTC)
The matter is utterly confusing. At the top of the page a trapezoid has two parallel sides. At the bottom said four sided figure has no tow sides parallel. Querobert August 21 2004
Is it just me, or does the second paragraph on the front page have the two terms swapped? We seem to have agreed on the following:
  • A quadrilateral with one pair of parallel sides is called a trapezoid in the US, and a trapezium in the UK;
  • A quadrilateral with no pair of parallel sides is called a trapezium in the US and a trapezoid in the UK.
However, the second paragraph of the article as it stands seems to directly contradict the second line above. Am I mistaken, or is it? After all, comments asserting the correctness of a paragraph can never themselves be wrong... Pmdboi 21:21, 17 January 2007 (UTC)
I went ahead and corrected it. Pmdboi 17:27, 18 January 2007 (UTC)
You are of course correct. A bad edit was made a week ago by an anon, who of course did not bother to satisfy my request for prior discussion. -- Meni Rosenfeld (talk) 17:20, 21 January 2007 (UTC)
Problem seems to have returned. The first and second paragraphs contradict each other on term/location. Wuzzled 11 September 2007 —Preceding unsigned comment added by Wuzzled (talk • contribs) 22:24, 11 September 2007 (UTC)
It is okay. Try reading the second paragraph again. -- Meni Rosenfeld (talk) 22:49, 11 September 2007 (UTC)

In the United States, a trapezoid has two parallel sides. — 131.230.133.185 10:30, 3 August 2005 (UTC)

(Edit by Arbiter) I'm changing the terms back to the way they're supposed to be according to every real dictionary... trapezoid is a British construction, trapezium is the American one (despite what you've heard in school). —Preceding unsigned comment added by 66.182.140.100 (talk) 15:44, 24 March 2008 (UTC)
66, your edit contradicts the dictionary reference at the bottom of the page. Do you have a reference supporting your edit? --Allen (talk) 16:05, 24 March 2008 (UTC)
In fact trapezoid is seldom used in the UK, and when it is used, it normally coincides with the American sense. dbfirs 22:46, 24 March 2008 (UTC)
  • I've reverted the edits by 66.182.140.100 who must be reading a confused dictionary. The word trapezoid almost exclusively and almost ubiquitously denotes a shape with two parallel sides, even in the UK where trapezium is more commonly used with this meaning for the quadrilateral. Both terms have been used in the past for a shape with no parallel sides, but it is a hundred years since this usage was common in the UK. Nearly all UK schools simply speak of an irregular quadrilateral, and have no special term corresponding to the American trapezium. The usage is confusing however you explain it! dbfirs 23:00, 24 March 2008 (UTC)

[edit] The school definition of T.

"A quadrilateral that has exactly two sides parallel." (bold is mine - GS), see f.e. http://www.math.com/school/glossary/defs/trapezoid.html.

I hate the definition because 1) I used to think a parallelogram is a trapezoid and 2) I know about open sets and closed sets and I beleve a practical definition should define a closed set, not an open one.

So my question to contributors: did you mean "exactly two sides parallel" or "at least two sides parallel"? and what should we do with all that? --GS 14:34, 18 Apr 2005 (UTC)

Indeed, in grade school (at least in Ontario, Canada), students are being taught that a trapezoid must have exactly two parallel sides; i.e., a parallelogram is not a trapezoid. This is inconsistent with other definitions that I have seen, so there certainly seems to be some controversy over the matter. In fact, the exactly-two-parallel-side definition seems to be the most prevalent (however silly that definition seems to me), so this Wikipedia article should probably be updated to at least mention this definition. --Pomakis 17:18, 29 January 2006 (UTC)

It means "at least" two sides parallel, to fit in the diagram which includes all quadrilaterals. "all Parallelograms are Trapeziums, all Trapeziums are Quadrilaterals" "some Quadrilaterals are Squares, all Squares are Parallelograms, which in turn, makes them Trapeziums" --Rkeysone 14 September 2006


The definition sounds a little mushy because it includes the following two contradictory lines:

"Some authors define it as a quadrilateral having exactly one set of parallel sides, so as to exclude parallelograms." "If the other set of opposite sides is also parallel, then the trapezoid is also a parallelogram" Seems some editing is it order. —Preceding unsigned comment added by 68.77.148.122 (talk) 12:40, 5 September 2007 (UTC)

"Some authors", not "all authors". You have also missed the line "[This article] admits parallelograms as special cases...". -- Meni Rosenfeld (talk) 12:48, 5 September 2007 (UTC)

[edit] Area

How come the area is (L1+L2)/2×H

Split the T. to two triangles, calculate and sum. What is your answer, anyway? --GS 14:18, 26 Apr 2005 (UTC)

[edit] Absolutely Merge it

goldenrowley 8-6-06

[edit] Different types of trapazoids

Requested: add different type of trapazoids. ex: isosceles trapazoid

Well, isoceles trapezoid is already discussed, what others do you have in mind? In either case, be bold! -- Meni Rosenfeld (talk) 16:30, 13 December 2006 (UTC)
Right trapezoid — Ti89TProgrammer 04:14, 10 October 2007 (UTC)

[edit] Definition of 'Midsegment'

Requested: please account for the definition of a midsegment in the case of parallelograms. The currently used definition of a trapezoid (i.e. a shape with two parallel sides) allows for the inclusion of parallelogram, which is fine. However, the definition of a midsegment then states that the midsegment is to be drawn from the midpoints of the non-parallel sides, which a parallelogram does not have. So, either a parallelogram doesn't have a midsegment (this, I think, is not the correct solution) or the definition of a trapezoid needs to be more restrictive to not include parallelograms (not everyone would be happy with that) or, and this is likely the best solution, the definition of a midsegment of a trapezoid needs to be modified to account for the case of parallelograms.

I've made a modification which aspires to solve this issue. -- Meni Rosenfeld (talk) 16:30, 13 December 2006 (UTC)

[edit] Circumscribed trapezoids / quadrilaterals

Maybe we should add some information about this too? --HappyCamper 19:59, 10 March 2007 (UTC)

[edit] Is parallelogram a trapezium(or trapeziod)?

Yes, I agree with the taxonomic classification of quadrilaterals illustrated. Parallelogram is just a special case of trapezium(or trapeziod). Let's consider the formulae used for parallelogram and trapezium(or trapeziod) in producing their area. The relationship in between them(the formulaes) agrees with the taxonomic classification of quadrilaterals illustrated. Thus, hereby i conclude that, PRACTICALLY, parallelogram is just a special case of trapezium(or trapeziod); it is a trapezium (or trapeziod).

    • by Sia S.H. 7th of April, 2007


I disagree with the classification above. There are no useful inherited properties in defining a parallelogram as a special case of trapezoids. The most notable disadvantage, however, is that there is now no word reserved for referring to a quadrilateral with exactly one pair of parallel sides. You cannot call it a "trapezoid" because some schmuck might say "well, technically, that could still be a parallelogram." Every textbook I have ever seen defines a trapezoid as a category exclusive of parallelograms.—Preceding unsigned comment added by Cyclehausen (talk • contribs)

That's like defining rectangles to exclude squares, or defining continuous functions to exclude differentiable functions. Such an approach is rare in mathematics, and its reasons for being commonly adopted in the case of trapezoids are traditional, not mathematical. -- Meni Rosenfeld (talk) 15:49, 30 June 2007 (UTC)

It really is not the same. When you talk about continuous functions, you are referring to a set of properties that they have that are useful. Differentiable functions inherit all of these useful properties (most notably the value of the two-sided limit at equalling the value of the function every point on the interval), and add a set of their own.

Squares inherit a number of useful properties from rectangles, and admitting squares as a special case of rectangles (and rhombi) allow them to inherit all the useful properties from each superset.

There ARE NO useful properties of trapezoids. They simply are (I attest) quadrilaterals with exactly one set of parallel sides. The only property that comes from this that MIGHT be considered a useful inheritance for parallelograms is the area formula, if you choose to see the parallelogram's area formula as a degeerate case of the trapezoid formula. And how useful is that, really? Yes, there is also the fact that the diagonals cut each other in the same ratio, but in a parallelogram the much more useful property that the diagonals bisect each other exists. Even the 4-side area formula for trapezoids fails to be valid when the shape is a parallelogram. The primary reason in all practical practice, both formally mathematical and secular, to use the term "trapezoid" is to DIFFERENTIATE it from a parallelogram.

Plese remain mindful that in Geometry, definitions have their greatest value when we use them to prove things. No proof of which I am aware for parallelograms utilizes any inherited properties of trapezoids that cannot be easily established from the definition of the parallelogram "A quadrilateral with two pairs of parallel sides" instead. That may seem obvious, and that is exactly why classifying a parallelogram as a trapezoid is unnecessary and confusing.

I do appreciate the counter-argument, but I must ask for some North-American textbook references that support it. Perhaps the American and British traditions of trapezoid warrant completely separate pages. Cyclehausen 11:04, 1 July 2007 (UTC)

I do not argue with exclusion of parallelograms being more common in the literature (though I have no references one way or the other). Consequentially, I have no objection to excluding parallelograms in the article, as Wikipedia is, after all, an encyclopedia. Regardless, though the points you make may be valid, I am not convinced that mathematically speaking, excluding parallelograms is superior. -- Meni Rosenfeld (talk) 11:31, 1 July 2007 (UTC)
There is one notable advantage you gain if you exclude parallelograms: a "pure trapezoid" is determined by the lengths of its sides. This is obviously not true for parallelograms. This is probably the reason why some people restrict the definition. I think that this fact should be mentioned on the page. —Preceding unsigned comment added by 147.175.96.224 (talk) 08:44, 30 April 2008 (UTC)
The glitch in the formula is mentioned at the appropriate point after the formula for area using just sides. Does anyone use this formula? Dbfirs 09:44, 1 May 2008 (UTC)

I really would be happy with a separate paragraph being devoted to the option to include or exculde, rather than a simple aside in the opening paragraph about it. I think the dichotomy is certainly notable. My main problem is that in Mu Alpha Theta (the high school mathematics competitive honor society) the definition I present is used and standardized, as it is in every high schoolt ext I have ever encountered, and students tend to try to dispute questions that implement our standard by referring to this article. It gets old.Cyclehausen 11:47, 1 July 2007 (UTC)

I suppose this can be a good change, but I am not currently inclined to do it myself. I will be more than happy to help if you decide to try this yourself. -- Meni Rosenfeld (talk) 16:42, 1 July 2007 (UTC)

[edit] A "pair of parallels" or a "set of parallels"?

Someone in hope to improve clarity changed "pair of parallels" sides of the trapezoid to "set of parallels" sides. I am not sure what "set" means in the geometry context. Isn't "pair" clear? Ricardo sandoval 03:28, 24 August 2007 (UTC)

Pair seems quite clear to me, and definitely better than "set". Pair is more specific, since a pair is a set with two elements. Doctormatt 18:34, 24 August 2007 (UTC)

[edit] "Exactly opposite"

The article now correctly gives the two terms for what North Americans call a trapezoid, but then says:

The exactly opposite concept, a quadrilateral that has no parallel sides, is referred to as a trapezium in North America, and as a trapezoid in Britain and elsewhere.

There is no "exactly opposite concept" to that of a quadrilateral that has parallel sides. The intended meaning is that the assignment of words to concepts is exactly opposite.

I suggest the phrasing:

Unfortunately, the same two terms are also used to refer to a quadrilateral with no parallel sides, in exactly the opposite manner: this is called a trapezium in North America and a trapezoid in Britain and elsewhere.

The article continues:

To avoid confusion, this article uses the North American wording.

This seems to imply that the North American wording, or rather terminology, is less confusing. The intended meaning must be that, to avoid confusion, the article uses only one convention, and it happens to be the North American one.

I suggest simply saying:

This article uses the North American term.

Normally I would just make these edits, but there is a warning saying not to touch this paragraph without first discussing it over here, so here I am.

--207.176.159.90 10:21, 1 September 2007 (UTC)

It's great that you have followed the request in the comment. In fact, it was placed there because of numerous people who were not aware of the alternative term, haven't bothered to actually read the paragraph, but only "knew" that trapezoid is NA and trapezium in UK, so they have switched the terms in the paragraph. Ironically, such people have continued to do so even after the comment was placed. Changes of phrasing are still welcome in the spirit of WP:BOLD.
As for the changes you suggest - I would say that, if it is understood that the object of our discussion is quadrilaterals, then "a quadrilateral with a pair of parallel sides" is the opposite of "a quadrilateral with no pair of parallel sides". I have added such a clarification, which I prefer to your suggestion since it emphasizes the contrast between the terms.
I would also say that the alleged implication of "to avoid confusion" is a bit of a strecth, but I see no harm in dropping that part, as you propose.
I have changed the article accordingly, see if it looks better to you. -- Meni Rosenfeld (talk) 12:42, 1 September 2007 (UTC)
I agree with 207; a quadrilateral with two parallel sides is not the "opposite" of one with no parallel side, and they certainly are not exactly opposite. It is the wording that is opposite. Meni, do you have a source that the two shapes themselves are considered "opposites" of each other? --Allen (talk) 16:01, 24 March 2008 (UTC)
I have made changes similar to what the original poster suggested. I also took out the giant "do not change without discussion" warnings on the sentence, which seem unnecessary to me. I also added "according to Merriam-Webster" to clarify why we're making this claim about the terms. I suspect the reason we've had trouble with people changing this is that Dbfirs is right when they imply above that the dictionary is wrong. (Dbfirs says, "The word trapezoid almost exclusively and almost ubiquitously denotes a shape with two parallel sides, even in the UK...") But as Dbfirs also seems to imply by upholding the current claim in the article, we have little choice but to go with our one so-called reliable source. --Allen (talk) 02:31, 27 March 2008 (UTC)
Oh yeah, and if we get a bunch of people switching the wording despite the changes I've made, I won't oppose putting the "do not change" note back in. --Allen (talk) 02:33, 27 March 2008 (UTC)
To be fair to Merriam-Webster, they were probably correct when the dictionary was compiled (100 years ago?), but I have collected a set of citations to prove my assertion that their claim (and that of other US websites) about British usage is no longer true. I did find one Wikipedian who was taught the old usage at school (by an old Maths teacher) so I can't claim that the alleged British meaning is totally obsolete, just that it is very rare in modern usage. The problem is that both words originally meant just a more general quadrilateral than a parallelogram. Distinguishing shapes with one pair of parallel sides is a distinction not considered by Euclid, and attempts to make the distinction have been confused (blame Proclus?). Perhaps the best place for my citations would be on this discussion page, rather than cluttering up the article? What does anyone else think? dbfirs 08:00, 27 March 2008 (UTC)

[edit] What about the other one?

In the last comment, I suggested "uses the North American term" because, after the discussion about the two versions of the terminology, the article turns out to be about trapezoids(NA) only. That's fine if that's what you're interested in, but what if you wanted to read about trapezoids(UK)? Looking under trapezium doesn't help: it's just a redirect to trapezoid. There seems to be no article about the trapezoid(UK), and I don't see why there shouldn't be one.

However, there isn't much to say about it, other than repeating the same content about the two conflicting usages. So what I suggest is that the two figures be treated as a single subject: although the usual rule in Wikipedia is one article per subject, I think it would make sense for this to be an exception.

So I propose that this article be renamed to something like "The trapezium and the trapezoid". Describe the two usages. Discuss the etymology (trapezium(UK) from the same root as "trapeze", as per its parallel sides; "-oid" because the pair of parallel sides wasn't there) and how the reversal of senses happened (one influential book, the OED says). The sort of thing that the article on long and short scales does. Then finally say "The rest of this article uses the North American terminology" and go on to discuss the trapezoid(NA) as now, and then briefly and the trapezium(NA). Howzat?

--207.176.159.90 10:34, 1 September 2007 (UTC)

That is, indeed, an interesting conundrum. I think the solution is: There is really nothing to be said about trapezium (NA), hence it is not notable enough to deserve an article. The article Trapezoid should use the NA wording, and mention trapezia (is that the correct plural?) only by virtue of their connection with trapezoids. In this context, mentioning the NA/UK confusion is appropriate, and explanations of etymology (about which I personally know nothing) would also be desirable. -- Meni Rosenfeld (talk) 12:53, 1 September 2007 (UTC)

[edit] "UK and elsewhere"

It's a little vague to lump everything together. Australians officially use trapezium for a quadrilateral with one set of parallel sides (as in UK), but also use trapezoid informally (as in NA) because of the influence of American educational shows. The quadrilateral with no parallel sides never had a special name - we always simply called it an irregular quadrilateral. Aspirex 10:23, 4 December 2007 (UTC)

We certainly cannot make any guarantees about how every person on earth chooses to call this. I think as long as a significant majority of non-American countries officially use the British convention (which is consistent with your statement), the claim in question is valid. -- Meni Rosenfeld (talk) 12:03, 4 December 2007 (UTC)
American "math" reference websites seem to believe that British people commonly use the word trapezoid to mean an irregular quadrilateral. This may have been the case a hundred years ago, but the word is rarely used in the UK now, and modern usage usually denotes either a British trapezium (= US trapezoid), or a 3-D shape having faces with some parallel sides. The statements in the article and on American websites are misleading about modern usage. dbfirs 10:41, 24 March 2008 (UTC)
Merriam-Webster gives a definition of a usage which was dying out in Britain a hundred years ago when this dictionary was printed (1913?) and is now virtually obsolete. Mathworld uses this out-of-date American view of British usage. In modern usage, trapezoid means the same almost everywhere, thus the article title is valid, and needs only the caveat that the article describes what is more usually called a trapezium in Britain and elsewhere. Perhaps it would be better to relegate the confusing American definition of trapezium to a footnote? In fact, both terms have been used, from Euclid onwards, to describe any general quadrilateral (more general than a parallelogram). The confusion arose from an ambiguous statement by Proclus.

The following show that modern non-American usage of the word trapezoid coincides with American usage.

Most of these citations are invalid. There is a difference in the UK between noun-usage (Trapezium) and adjectival usage (Trapezoid or less commonly Trapezoidal). In the same way that no one would say "Hexagon table" (correct usage is "Hexagonal"), neither would anyone say "Trapezium table". —Preceding unsigned comment added by 88.105.28.221 (talk) 09:17, 18 April 2008 (UTC)
Fair comment, but the adjectival use derives from the noun definition, and thus serves to illustrate current usage.
-- or are you claiming that trapezoid is now just the adjective from the noun trapezium? If so, then my case is proven! Dbfirs (talk) 07:18, 21 April 2008 (UTC)


[edit] Contradictory statement

The sentence which begins "The exactly opposite kind of quadrilateral ..." obviously contradicts the previous paragraph. Along with other editors, I have not corrected it because of the request to discuss it here first, but it has long needed correction. I hereby give notice that I intend to correct this statement, (also removing the claim that the word trapezoid is regularly used in the UK). The whole area is a minefield because of contradictory usage, so I shall edeavour to retain a neutral point of view, retaining the US usage since that was the original article. Is this OK? dbfirs 10:07, 17 January 2008 (UTC)

this is the correct version. I don't know if you were referring to this or the wrong version. Note that countless hours of work have been dedicated to making sure this paragraph is correct, and except for brief periods of time after some smart-ass tries to change it, it is perfectly valid. -- Meni Rosenfeld (talk) 11:30, 17 January 2008 (UTC)
By the way, while the article was in the wrong version at the time you wrote this post, you say that "it has long needed correction", which suggests you have seen its correct version. In that case, references exist for a reason. -- Meni Rosenfeld (talk) 11:37, 17 January 2008 (UTC)
Apologies to Meni Rosenfeld. Somehow I missed picking up the fact that the erroneous version had only been there for eleven hours. I had looked at a version from November which had the same error, and assumed (wrongly) that the error had been there since then. If I had looked more carefully at the history I would simply have reverted the vandalism and left it at that. I can see that many hours of work have gone into ensuring the accuracy of the article, so I will not make any changes, except to comment here that Euclid originally used "trapezium" to mean a general convex quadrilateral not necessarily having any parallel sides (but including all others quadrilaterals), then from about 1700, the word began to be used in England for the quadrilateral with one pair of parallel sides, and this usage gradually became dominant in the UK. Use of the term "trapezoid" is rare in the UK, but it is normally used in the US sense when it does appear. The last use of "trapezoid" to mean a general convex quadrilateral with no parallel sides was by R F Burton in 1851 (as far as I can determine). Perhaps someone can correct me if I am wrong.
Because current British and Commonwealth usage has "trapezoid" in the US sense (as a synonym for the UK "trapezium"), I feel that a slight adjustment to the article would help to clear up continuing confusion. Again, apologies to Meni Rosenfeld who obviously feels deeply about this subject. dbfirs 14:04, 17 January 2008 (UTC)
P.S. Would you like some references to contradict your fallacious reference? Trapezoid definition (Only the table is wrong!) dbfirs 14:11, 17 January 2008 (UTC)
Huh? Everything on that page (which is in any case less reliable then mine) agrees with the article, so I'm not sure what your point is. This leads me to wonder which parts, if any, of your post are sarcastic. If you can find a reliable reference that contradicts the article you can edit accordingly.
I don't feel deeply about this subject. I only feel deeply about people who don't bother to think half a second before making nonsensical edits (like the recent one by anon). -- Meni Rosenfeld (talk) 15:40, 17 January 2008 (UTC)
Sorry, I'm not communicating very clearly, am I? I was apologising for my error in suggesting that the reversed terms had been there for some time, when in fact it was just vandalism or thoughtless editing that had appeared repeatedly. You corrected the vandalism and I am happy with that, and I agree with your opinion of those who make alterations without thinking. My only query was over British usage of "trapezoid". In fact, nearly all British and Commonwealth usage is the same as the US meaning. The table which you quote as reference suggests that the usage is the opposite in the UK for "trapezoid", when in fact it is only "trapezium" which has a different meaning. I will collect together some reliable references (I agree the "mathopenref" is less reliable than your page). Best wishes, and I am not arguing with you or being sarcastic, just trying to express the truth as clearly as possible, which I hope is what all true Wikipedians aim to do. dbfirs 17:49, 17 January 2008 (UTC)
Okay. You are no doubt familiar with Wikipedia:Verifiability, so finding reliable sources to back up your claim is crucial. -- Meni Rosenfeld (talk) 17:58, 17 January 2008 (UTC)
Thanks, I'll take care. By the way, from your knowledge of US mathematics, do you know whether the US term "trapezium" normally includes concave quadrilaterals, or is it reserved for just the convex variety which do not satisfy other criteria? (Genuine question, not trying to catch you out or anything!) dbfirs 19:38, 17 January 2008 (UTC)
I actually never claimed to have any knowledge of US mathematics, and indeed I have none. According to definitions such as the Merriam-Webster one, it seems to include concave quadrilaterals as well, but your guess is as good as mine. -- Meni Rosenfeld (talk) 19:47, 17 January 2008 (UTC)
Thanks, and sorry again for jumping to conclusions (based solely on your use of US sources and references). I'll leave you in peace to get on with valuable editing. dbfirs 20:25, 17 January 2008 (UTC)

[edit] Set or pair?

Why did we change pair to set? I always thought that a pair was a set of two, and thus pair would give a more precise definition. Have I missed some subtlety? dbfirs 08:28, 14 April 2008 (UTC)

Since no-one has objected, and two other editors have agreed in the past, I am making the change. Dbfirs (talk) 07:24, 21 April 2008 (UTC)