This thread here reminded me of a stupid argument I had with a friend recently. Perhaps someone can give me the definitive answer and prove me right. 
Metres squared vs square metres. The same thing or different? If so, how so? (Americans please feel free to substitute ‘feet’ for ‘metres’ if it helps.)
It’s the same for 1 meter.
“My garden is four square meters” = “My garden has an area equal to four plots, each one meter by one meter” (my garden could be two meters by two meters, for example).
“My garden is four meters square” = “My garden measures four meters by four meters” or sixteen square meters total.
If I had a nickel for every time I’ve seen this misused (by journalists who are paid to write factually, for heaven’s sake), leading to ridiculous statements (like, the Metro Detroit area is over a thousand miles square), I could at least buy a cup of coffee.
zut, I’m very glad that this bothers other people as well. I cam ontinually astounded by news reports that contain contradictory numbers in two consecutive sentences because journalists can’t understand what numbers actually mean..
And you have the difference between meters squared and square meters correct. They are entirely different except, as starfish points out, for the special case of one meter.
Futile Gesture, I sure hope you were on the right side of this argument.
It’s not really a special case.
One square meter is the number of square meters contained in a one-square-meter area.
One meter square(d) is the number of square meters contained in an area one meter on a side.
Coincidentally, since 1*1=1, they are the same number. If one substitutes “two” for “one” above, pluralizing as necessary, the italicized “One” becomes “Four.” But with those alterations, the statements remain true.
Perhaps the most famous example, the one with the greatest practical impact, comes from Article I, Section 8, Clause 18 of the U. S. Constitution:
This version of the Constitution is maintained by the U.S. House of Representatives and can be found at this link.
As I recall, the actual District, known as the “District of Colombia,” is only 64 square-miles in area, but if you look at a map, I think that you can see the outlines of the old 10-miles-on-a-side square. I don’t know what happened to the rest of the square. 
Thanks for the replies. Yes, I win this argument.
A big clue seems to be, to me, related to the square/squared issue. Looking at this problem from a linguistic position I would adjudge that saying ‘squared’ would indicate that the figure quoted requires to be multiplied with itself (i.e. squared) before reaching the desired bottom line. But saying ‘square’ indicates that this has already been done.
Thus 4 metres squared = 4²m = 16 square metres = 16m².
So when people drop the ‘d’, as zut did, it confuses things, as people have to rely on the order the words appear which is exactly the opposite of how they are usually written.
All in all it’s a bit overly confusing. Did no-one think this through at the start?
The confusion stems from the fact that a square-meter (Sorry for the US spelling, but what can I do? I’m a Yank.) is a unit of area in its own right. People often leave off the hyphen that makes it a compound word (not that that’s incorrect, it’s just confusing to people who aren’t used to thinking about units).
We should come up with some other name for this unit, which will remove confusion. You know, the way that they call a newton-meter of work a joule, so that people don’t think it’s a unit of torque. Since it was my idea, I say that the square-meter be henceforth known as the Saltire. From now on, a plot 100 meters on a side will be said to have an area of 10 kS (kilosaltires). (Of course, that’ll just make it hard on the heraldry crowd, but there are fewer heralds than there are journalists.)
Too late. It’s a hectare.
Look at Arlington, VA. The land south of the Potomac was ceded back to Virginia by the Feds. prior to the Civil War, for reasons I’m not aware of.
You’re right, of course. Oh, well. So we need to tell the journalists to describe areas in hectares. If they’re talking about something that’s 1 square-meter in area, they’ll have to call it 100 millihectares. Or a tenth of a centihectare.
I still say it would be better to have a separate name for a square meter, rather than a name for 10,000 of 'em. But I guess the ISO doesn’t care what I think.
I’m such an idiot. I meant to say 100 microhectares or a tenth of a millihectare.
At least, I think that’s right.
Sorry, but I’m going to have to side with Futile’s friend.
In the OP, the question was the difference between “meters squared” and “square meters.” In my educational experience (BS in electrical engineering, several grad. classes) these terms have been interchangable. Consider that the term “4m” is read “four meters.” That is, the number “four” followed by the unit “meter(s).” The term “4m^2” is then read “four meters squared” - the number “four” followed by the unit “meters squared.” The “d” in “squared” is vital.
I agree with zut, but notice that his post left off the “d”. Saying “four meters square” is similar to saying “four meters long.” Here, “square” is an adjective meaning that the area happens to be a square. If you say “squared” you are reffering to the unit, not the number.
As long as newspapers don’t start selling advertising space by the nanohectare, everything is alright.
An area 10 m on each side is called an are. An area 1 m on each side is called a * centare* or a centiare.
Calling a unit a “square-meter” is silly and German-sounding. The proper unit is meters-squared.
Then we’ll have one problem.
1 m[sup]2[/sup] is the standard SI unit of area. If you, however, derive it from are by putting a prefix in front of it, are would be the standard unit and square-meter/meter squared (call it whatever you like) would be a derivation of the standard, while in fact it is the other way round.
This can be very nasty in physics, when you use different units in formulas. The most convenient thing is: If you give everything in standard units, the result will be in a standard unit, no need to convert.
(This very mess occured with mass. The SI unit of mass is the kilogram, although it seems to be derived from the gram. In fact, however, the gram was defined as 1/1000 kilogram and not, as one could assume, the kilogram as 1000 grams.)
If you ask me, I’m still waiting for the day when we count time in kiloseconds.
You’ll notice that I didn’t define m[sup]2[/sup] in terms of ares; I merely pointed out that there’s no such thing as a “centihectare.”
A centihectare wouldn’t make sense either. The prefix hect- means 100, and centi- means 1/100, so they cancel each other out.
“An area of four meters squared” refers to any area having sixteen square meters. “An area of four meters square” refers to, specifically, an area that measures four meters by four meters. From www.m-w.com: square [2 adjective] 4b : being of a specified length in each of two equal dimensions <10 feet square>
I guess I’ve usually seen “meters square” misused more often than “meters squared”, which is why I replied as I did before.
(apologies if this double-posted)
It’s my position that “four meters squared” is ambiguous, and can mean either an area of (4 m)[sup]2[/sup] or 4 m[sup]2[/sup].
I agree with zut’s point that “x meters square” refers definitely to a square that is x meters on a side. And square meters is similarly clear. But in my experience, it is quite often the case that “x meters squared” is used to refer to x m[sup]2[/sup].
Most of the time the meaning is clear from context. For example, if someone writes, “168 m[sup]2[/sup]” and then refers to it as “one hundred sixty-eight meters squared”, it’s not confusing. Written out in words, though, it isn’t clear. I suppose in terms of the bet this means that there is a difference, but one could mean the same thing, depending on context. Regardless, it’s better to avoid ambiguity and use some clear form (like “one hundred sixty-eight square meters”).
Here’s a link to someone saying basically what I just said : Ask Dr. Math : Meters Squared vs. Square Meters
The response quotes from the NIST stylistic conventions (which kind of skirt the issue a little bit), although his(?) quote leaves out a line that supports the use of squared to apply to units :