Talk:Trapezoid

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

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

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)

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)

Absolutely Merge it

goldenrowley 8-6-06

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)

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)

Circumscribed trapezoids / quadrilaterals

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

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)

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)

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)

"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)

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)