What type of quadrilateral is this? (not "trapezium". maybe "trapezoid"...? or just "irregular"?)

[chijioke]

I know that that this is not a trapezium because a trapezium has only two pair of opposite sides parallel but in this case none of the sides are parallel talk more of even being equal?

 

[topsquark]

I know that that this is not a trapezium because a trapezium has only two pair of opposite sides parallel but in this case none of the sides are parallel talk more of even being equal?


View attachment 36183

If none of the sides are equal, it's simply called a quadrilateral. Or, perhaps, an irregular quadrilateral.

-Dan

 

[chijioke]

Is there any special properties for irregular quadrilaterals or do they share the same general properties of regular quadrilaterals? E.g sum of interior angles of quadrilateral= 360 degrees

 

[topsquark]

Is there any special properties for irregular quadrilaterals or do they share the same general properties of regular quadrilaterals? E.g sum of interior angles of quadrilateral= 360 degrees

Pretty much the only property that an irregular quadrilateral shares with any other quadrilateral is that the sum of the interior angles is 360 degrees. This holds true for even one that is concave.

Well, that and they all have four sides!

-Dan

Addendum: Since I mentioned that, your quadrilateral is also convex. So you could possibly call it a convex irregular quadrilateral.

 

[The Highlander]

I know that that this is not a trapezium because a trapezium has only two pair of opposite sides parallel but in this case none of the sides are parallel talk more of even being equal?

That is called a Trapezoid (Trapezium like, but with no sides parallel) except in the Americas (where they might call it a Trapezium? Very confusing, I know. ??‍). See (highlighted) sections here.

 

[topsquark]

That is called a Trapezoid (Trapezium like, but with no sides parallel) except in the Americas (where they might call it a Trapezium? Very confusing, I know. ??‍). See (highlighted) sections here.

I learned the definition that chijioke quoted, about needing to have two parallel sides to be called a trapezium.

And, yes, I'm from upstate New York!

-Dan

 

[The Highlander]

I learned the definition that chijioke quoted, about needing to have two parallel sides to be called a trapezium.

And, yes, I'm from upstate New York!

-Dan

Doesn't that put you rather 'at odds' with the majority of your countrymen (& women ?)?

This discrepancy only became apparent to me when I joined this forum ("Britain and America are two nations separated by a common language"! ?) .

If you search the forum for "trapezoid" (ticking "Search titles only") the first two results verify the differing usage...
See here, where the OP's use of "math" (rather than "maths") suggests (to me) s/he's on your continent.
and
See, also, here, where the incontestable @Dr.Peterson himself actually provided the definition!
(You even 'liked' that one. lol.)

 

[topsquark]

Doesn't that put you rather 'at odds' with the majority of your countrymen (& women ?)?

This discrepancy only became apparent to me when I joined this forum ("Britain and America are two nations separated by a common language"! ?) .

If you search the forum for "trapezoid" (ticking "Search titles only") the first two results verify the differing usage...
See here, where the OP's use of "math" (rather than "maths") suggests (to me) s/he's on your continent.
and
See, also, here, where the incontestable @Dr.Peterson himself actually provided the definition!
(You even 'liked' that one. lol.)

A definition is a definition. I've never really liked the word "trapezium" in any event.

I'm a QFT Physicist. Let me be weird!

-Dan

 

[The Highlander]

A definition is a definition. I've never really liked the word "trapezium" in any event.

I'm a QFT Physicist. Let me be weird!

-Dan

In that case I expect you're perfectly comfortable with it being both definitions at the same time! ?

 

[topsquark]

In that case I expect you're perfectly comfortable with it being both definitions at the same time! ?

I believe in three impossible things before br(secondbreakfast)unch.

-Dan

 

[Dr.Peterson]

I learned the definition that chijioke quoted, about needing to have two parallel sides to be called a trapezium.

And, yes, I'm from upstate New York!

-Dan

That is peculiar. I probably didn't hear that usage until I started communicating internationally, and I'm from Connecticut and live in upstate NY.

On the other hand, I've long felt that giving "trapezium" the more restrictive definition made more sense logically, since "-oid" means "sort of like". On the other other hand, etymologically neither is obvious, as "trapezium" comes from Greek for "table".

Checking the history, I find that Miller says,

TRAPEZIUM and TRAPEZOID. The early editions of Euclid 1482-1516 have the Arabic helmariphe; trapezium is in the Basle edition of 1546.

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).

Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid.

So the American usage was invented in the UK, but became popular only (mostly?) over here. It almost sounds like a printer's accident. But Americans can't be blamed, apart from our attraction to the latest thing.

I know that that this is not a trapezium because a trapezium has only two pair of opposite sides parallel but in this case none of the sides are parallel talk more of even being equal?


View attachment 36183

My first thought when I saw the question was, how about "scalene quadrilateral" (which I find is used somewhere, but not as clearly defined as I expected). I didn't even think of "trapezoid" (British) or "trapezium" (US).

Is there any special properties for irregular quadrilaterals or do they share the same general properties of regular quadrilaterals? E.g sum of interior angles of quadrilateral= 360 degrees

Since, whatever you choose to call it, you are defining it negatively (by what it is not), it has no (positive) properties of its own; it has all the properties of any quadrilateral, and none of those that are reserved for special kinds.

 

[The Highlander]

So the American usage was invented(?) in the UK, but became popular only (mostly?) over here.

Almost sounds like your blaming us! ?

It almost sounds like a printer's accident.

My original link did refer to it as a "mistake" (later corrected by us).. ?

But Americans can't be blamed, apart from our attraction to the latest thing.

Maybe it was the only Maths book in the Mayflower's library? ?
(I know that's anachronistic; it was only a joke.)

If anyone's to "blame" (other than Hutton himself or his publisher) it would be the US Army! ?️?
(As indicated here. ?‍)

 

[Dr.Peterson]

My original link did refer to it as a "mistake" (later corrected by us).. ?

I'd failed to read the Wikipedia article. So I see the British did go along with it for a while, along with the Americans. It doesn't say who rescued you after the 80 years.

 

[The Highlander]

I'd failed to read the Wikipedia article. So I see the British did go along with it for a while, along with the Americans. It doesn't say who rescued you after the 80 years.

Probably a respected authority like your good self who pointed out that it was much more logical to have 'Trapezium' be the "restrictive definition" (two sides being parallel a requirement) since 'Trapezoid' essentially means like a Trapezium (but not one!). ?

No doubt several people noticed that Hutton's book (having become so "popular") contradicted 13 centuries of accepted Geometrical terminology. It perhaps just took the 80 years to convince everyone who had 'invested' in Hutton's book that the mistake in it should be ignored?
(Of course, Hutton himself (and/or his publisher) may even have recognized the error but there being no Internet in those days, promulgation of errata may well have taken 8 decades to reach all points! ?‍?)

It's perhaps also worth pointing out that (as far as I can ascertain; I have no definite proof) it's not just the UK that uses the original (Proclus') definition. It's used everywhere other than in the 'New World' and. perhaps, only in North America at that. ?
(Possibly due to Hutton's book never having been translated into any other languages than English?)

 

[The Highlander]

It may also be worth mentioning that around the time of Hutton's publication, we weren't long out of a war with you guys! Maybe, having noticed the gaffe this side of the pond, the sentiment was: "Just let 'em wallow over there." ?

 

[Bruce]

Maybe, having noticed the gaffe this side of the pond, the sentiment was: "Just let 'em wallow over there."

Maybe. There must've been more than enough wallowing to deal with on their own side, at that point.

 

[khansaheb]

Much ado about nothing........