# Is a square a trapezium?

The only (British) definition for a trapezium I can find is "a quadrilateral with two sides
parallel".

It doesn't stipulate either way whether this means exactly two parallel sides or at least two sides.

Any ideas?

Paul
"A quadrilateral having no parallel sides. "

from www.dictionary.com

Since a square does have parallel sides it cannot be a trapezium, according to this definition.

--
MESSAGE ENDS. John Porcella

"tequilla2k" <[email protected]> wrote in message news:[email protected]...
[q1]> The only (British) definition for a trapezium I can find is "a[/q1]
[q1]> with two sides parallel".[/q1]
[q1]>[/q1]
[q1]> It doesn't stipulate either way whether this means exactly two parallel sides or at least[/q1]
[q1]> two sides.[/q1]
[q1]>[/q1]
[q1]> Any ideas?[/q1]
[q1]>[/q1]
[q1]> Paul[/q1]
"tequilla2k" <[email protected]> wrote in message news:[email protected]...
[q1]> The only (British) definition for a trapezium I can find is "a[/q1]
[q1]> with two sides parallel".[/q1]
[q1]>[/q1]
[q1]> It doesn't stipulate either way whether this means exactly two parallel sides or at least[/q1]
[q1]> two sides.[/q1]

You should know I am in the US, so assign credibility of my opinion (or lack thereof!) accordingly.

http://mathworld.wolfram.com/Trapezium.html

...says exactly the same thing so perhaps is the very definition you are referencing. It also says
the definition is equivalent to that for a trapezoid, and if you follow the link for trapezoid the
definition is the same (athough the diagram seems to suggest something, it doesn't explicitely rule
out the other possibility.)

If you look at:

...this *seems* to explicitly suggest that a parallelogram (hence a square) is a special case of
trapezoid. But if you read the detailed footnote, we see that unfortunately even in the US the
meaning of "trapezoid" (hence the meaning of the British trapezium) is not completely agreed upon.
On one hand a parallelogram qualifies as a trapezoid, on the other hand it does not.

This only adds to the confusion. We are saying the British trapezium is the same as the American
trapezoid, while also saying the American trapezoid has either of the two definitons you list above!
So, apparantly my reply serves no useful purpose other than to demonstrate the fact that we are more
confused than you are!

--
Darrell
Interestingly, my old Penguin Dictionary defines it as a "quadrilateral...with only two
parallel sides."

How can the definitions be so different?

--
MESSAGE ENDS. John Porcella

"John Porcella" <[email protected]> wrote in message
news:[email protected]...
[q1]> "A quadrilateral having no parallel sides. "[/q1]
[q1]>[/q1]
[q1]> from www.dictionary.com[/q1]
[q1]>[/q1]
[q1]> Since a square does have parallel sides it cannot be a trapezium,[/q1]
according
[q1]> to this definition.[/q1]
[q1]>[/q1]
[q1]>[/q1]
[q1]> --[/q1]
[q1]> MESSAGE ENDS. John Porcella[/q1]
[q1]>[/q1]
[q1]> "tequilla2k" <[email protected]> wrote in message news:[email protected]...[/q1]
[q2]> > The only (British) definition for a trapezium I can find is "a[/q2]
[q2]> > with two sides parallel".[/q2]
[q2]> >[/q2]
[q2]> > It doesn't stipulate either way whether this means exactly two parallel sides or at least two[/q2]
[q2]> > sides.[/q2]
[q2]> >[/q2]
[q2]> > Any ideas?[/q2]
[q2]> >[/q2]
[q2]> > Paul[/q2]
[q2]> >[/q2]
[q1]>[/q1]
Thats the American definition. A British trapezium does have parallel sides.

But is it exactly two or at least two.

"John Porcella" <[email protected]> wrote in message
news:[email protected]...
[q1]> "A quadrilateral having no parallel sides. "[/q1]
[q1]>[/q1]
[q1]> from www.dictionary.com[/q1]
[q1]>[/q1]
[q1]> Since a square does have parallel sides it cannot be a trapezium,[/q1]
according
[q1]> to this definition.[/q1]
John Porcella <[email protected]> wrote:
[q1]>"A quadrilateral having no parallel sides. " from www.dictionary.com Since a square does have[/q1]
[q1]>parallel sides it cannot be a trapezium, according to this definition.[/q1]

According to Chambers, in US usage a trapezium is a quadrilateral with no two sides parallel, in UK
it is a quadrilateral with exactly two sides parallel.

Weird, eh?
--
Rob. http://www.mis.coventry.ac.uk/~mtx014/

"John Porcella" <[email protected]> wrote in message
news:[email protected]...
[q1]> Interestingly, my old Penguin Dictionary defines it as a "quadrilateral...with only two[/q1]
[q1]> parallel sides."[/q1]
[q1]>[/q1]
[q1]> How can the definitions be so different?[/q1]
[q1]>[/q1]
[q1]>[/q1]
[q1]> --[/q1]
[q1]> MESSAGE ENDS. John Porcella[/q1]
"tequilla2k" <[email protected]> wrote in message news:[email protected]...
[q1]> The only (British) definition for a trapezium I can find is "a[/q1]
[q1]> with two sides parallel".[/q1]
[q1]>[/q1]
[q1]> It doesn't stipulate either way whether this means exactly two parallel sides or at least[/q1]
[q1]> two sides.[/q1]
[q1]>[/q1]
[q1]> Any ideas?[/q1]
[q1]>[/q1]
[q1]> Paul[/q1]
[q1]>[/q1]
[q1]>[/q1]
From "A School Geometry" circa 1935. (British) 'A Trapezium is a quadrilateral with one pair of
opposite sides parallel' Rog
Is a square a type of trapezium?

I'm searching around in the depths of my brain here but I'll have a go (sorry for any errors!) ...

A quadrilateral = a 2D shape with 4 sides.

A trapezium = a quadrilateral with *at least one pair* of parallel sides

A parallelogram = a quadrilateral with two pairs of parallel sides

A rectangle = a parallelogram with 4 right angles

A rhombus = a parallelogram with 4 equal sides

A square = a rhombus with 4 right angles.

Note: these are definitions using everyday terms as i recall them!

So ...

a square is a special rhombus,

but a rhombus is a special parallelogram therefore a square is a special parallelogram

but a parallelogram is a special trapezium therefore a rhombus is a special trapezium

but since a square is a special rhombus which is in fact just a special trapezium it (hopefully)
follows that a square is just a special trapezium.

I seem to recall a poster from when I was at school which had the various quadrilaterals as subsets
of eachother .... at least that's what I am basing my answer on.

I prepare to be corrected!

Steve
A trapezium is a trapezoid
A trapezium is a trapezoid, therefore it cannot be a square. A square has four congruent sides and angles. The most congruent sides a trapezium can have is three. The most congruent angles it can have is two.
A trapezium is a quaderalateral, so is a square
a trapezium has a top smaller than bottom or vice versa
I found this helpful:

Trapezium/Trapezoid

From John Conway: For many centuries, the word "trapezoid" was used to describe a four sided shape with no two sides parallel, while "trapezium" was used for one with two parallel sides.
Unfortunately, around 1800 they were used with the reverse definitions in a book by Charles Hutton that was so influential that this reverse usage rapidly became common. By about 1875 the English geometers managed to switch back, but the American ones never did, with the result that by about 1900 the two terms had opposite meanings on the two sides of the Atlantic.

Some dictionaries will tell you this is still the case, but what has actually happened is both words now refer to the one with two parallel sides "trapezoid" being the preferred term in America and "trapezium" in England. So a new word was needed for the general quadrilateral that might have no pairs of parallel sides and "quadrilateral" was resuscitated for this purpose.

Trapezoid
A quadrilateral having one pair of opposite sides parallel and unequal. From the Latin: trapezium a small table.
I'm a secondary teacher and the Year 7 and 8 optional tests have just such a question - there's a picture of a square and you have to tick is it a ... square, quadrilateral, trapezium and/or parallelogram (you are told to tick more than one)
You only get the marks if you don't tick trapezium - because it is all the others - and trapezium is exactly one pair of parallel sides, not two.
Since no-one's mentioned it: a square is not only a special kind of rectangle, rhombus, and parallelogram, it's also a special kind of kite. Of the best-known types of quadrilateral (square, rhombus, rectangle, parallelogram, kite, trapezium), the trapezium is the odd one out in that no trapezium is ever in any of the other categories.

Dave
In my search for the truth on this matter, I read this, supposedly Euclid's definiton of quadrilaterals:

[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI22.html]

"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 its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. "

So ... this suggests a trapezium could have one or even no pairs of parallel sides