The Logic Behind Metric Paper Sizes

Oily Pakora writes "Those of us in the United States are so used to our Letter and Legal paper sizes. We've seen the A4 paper size option in our printer trays and in printer preference menus. Metric sizes used almost everywhere in the world, save for the US and Canada. Here is an interesting article that discusses all of the aspects of metric paper. For those who enjoy a bit of math, did you know that in the Metric paper system, the height-to-width ratio of all pages is the square root of 2? This means that you can place two sheets of A4 side-by-side and they will equal an A3 sheet exactly, and two sheets of A3 will equal an A2."

2 x A4 = A3

[by Anonymous Coward]

News?

Re:2 x A4 = A3

[Leffe (686621)]

Not at all, I thought everyone knew this... I don't live in America though...

Re:2 x A4 = A3

[boaworm (180781)]

This whole thing reminds me of a thing Jerry Seinfeld did, making fun of chinese people eating with sticks. He said something about that he could not understand why they kept eating with sticks, since "they have seen the fork".

It feels just like that hearing about "US Letter", Yards, Pounds, Stones, Miles et al. YOU HAVE SEEN THE METRIC SYSTEM ;-)!

(btw, thanks France!)

Re:2 x A4 = A3

[hpa (7948)]

A0 paper is one square meter.

A4 paper is 2^-4 = 1/16 square meter.

Re:2 x A4 = A3

[chris_mahan (256577)]

a square meter. a square meter is the base of a cube meter. The cube meter is the volume of one ton of fresh water at sea level at the equator at zero degrees Celcius. 1 ton is 1000 kilo gram. each kilo gram is thus 10cm*10cm*10cm, which happens to also be a liter. 1 gram is 1 millionth of a ton, of 1cm*1cm*1cm. so if a bottle of water is 1000 grams (1 kilo gram), it is also 1 liter. So now I know the volume, the weight, and the measurement of the container. Pretty nifty no?
Density is expressed in a ratio from fresh water at zero degrees at sea level at the equator. Let's say the density of velveta cheese is 1.001. With this, I could tell you the size of a kilo of velveeta, and how large a container to use, and thus how much paper to use to wrap it in. Then I could express this in how many per A0, A1, or A2, since they are derived from the meter. Get it?

Class dismissed.

It's articles like this

[by Anonymous Coward]

That make me happy my rolling paper is not metric
-B

Re:It's articles like this

[justforaday (560408)]

That make me happy my rolling paper is not metric

oh, i dunno...can't say i'd mind something rolled with a sheet of A0... @_@

Side-by-sideness

[TyrelHaveman (159881)]

You can also put two 8.5x11 (Letter) sheets of paper side by side and it equals an 11x17 (Tabloid) sheet of paper...

Re:Side-by-sideness

[SSpade (549608)]

But 11x17 is not the same shape as 8 1/2x11.

That's the real beauty of A4/A3 etc. All the sizes in a given series (A00, A0, A1, A2, A3, A4, A5... or B1, B2, B3...) are the same shape.

So you can photocopy an A4 document onto A3 paper expanding it by the right proportion and it'll fit perfectly. And you can copy two A4 documents onto A3 paper and it'll fit perfectly. Or use psnup to put A4 formatted documents reduced to 2-up on A4 paper with no wasted space.

Try that with letter or legal size....

Re:Side-by-sideness

[srussell (39342)]

Yes, but the metric system is designed so that this ratio applies all the way from the smallest size up to the largest. So, two A4 is one A3, two A3 is one A2, two A2 is one A1.

Putting them side-by-side isn't as interesting as cutting them in half, though. I discovered this when I started printing photos from my inkjet. Photo paper is generally available in the stores in limited sizes. I can buy a bunch of A4, and cut it in half, and I have two A5s. Do it again, and I have A6s, which is nearly a 4x6. Best of all, these paper sizes are all standard, which is good, because my printer doesn't like me to define my own paper sizes. With the American system, I have to measure and cut, which is more difficult.

When you discuss the advantages of metric, it really is about convenience[1]. There's nothing that you can do with metric that you can't with the English system; it is just, generally, more difficult to do with the English system. If you don't care about convenience, and you live in the USA, then you probably don't have any reason to use metric.

[1] Of course, using metric in the US imposes a certain amount of inconvenience from compatability issues, but that's another argument.

Re:Side-by-sideness

[barawn (25691)]

When you discuss the advantages of metric, it really is about convenience[1]. There's nothing that you can do with metric that you can't with the English system; it is just, generally, more difficult to do with the English system. If you don't care about convenience, and you live in the USA, then you probably don't have any reason to use metric.

That's not quite true - one of the reasons that the Imperial system is moderately convenient for building is that base 12 is divisible by 2,3,4 and 6, so you'll encounter less rounding error if you need to split things up into common numbers. Base 10 is only divisible by 2 and 5. (Incidentally, this is of course why one of the older civilizations used base 60 - it's divisible by 2, 3, 4, 5, and 6, and it's the reason we have 60 seconds in a minute, 60 minutes in an hour).

So, for instance, if you want to break a 1' object into thirds, you can do it exactly. Try doing it with meters - it's 33 and a third centimeters. Most people would say "screw it, it's 333 mm" - but if you now take those "1/3 m" sticks and put 300 of them end to end, you don't have 100 m - you have 99.9 m, and you're a full ten centimeters short. In imperial, 1/3 of a yard is 1 foot. No rounding errors.

There really *are* advantages to the Imperial system - most people, however, simply assume that Imperial sucks and leave it at that.

Metric paper, however, is better designed than US. Being able to print 2 A4 on 1 without much work really kicks.

Re:Side-by-sideness

[ajs (35943)]

Also, if you take 3 8.5x11 sheets, line them up along their longer sides, attach them to eachother, put a staple through the middle of the first and second sheet join and then hang them from a height of approximately eye-level it makes the idea place for a picture of a naked "girl next door".

My buddy Heff taught me that trick.

Obligatory

[swordboy (472941)]

PC Load Letter!? WTF does that mean?!

Audi A4

[by Anonymous Coward]

There's also an Audi A4, and if you put two of those side by side, people say "Look, isn't that a coincidence".

Re:Audi A4

[Golias (176380)]

Two Mini Coopers side by side == One Audi A4
Two Audis == One BAM ("Big Assed Mercedes")
Two Mercedes == One average European house.

Wow, those Europeans can apply simple metric system math to everything!

Meanwhile, in America:
Two Mini Coopers side by side == One speed bump for a Hummer H2.
Two Audis in the driveway == A good house to break into.
Two Mercedes == Really, really tacky.
Two Hummers == The energy consumption of a typical third-world country
Two thrid world countries == A re-unified Germany. (I keed!)

The Metric System Sucks!!

[USAPatriot (730422)]

I am of the firm belief that the metric system sucks. It is a global conspiracy created to cause the downfall of all things that we know and love. Upon careful examination it is clear that the metric system is at least indirectly responsible for most of the world's problems, including but not limited to:

* Government conspiracy
* Microsoft Windows
* Rap Music
* Hondas and their drivers
* Transistors
* Pokemon
* Jerry Springer
* Televangelism
* Toxic waste
* The Republicans
* The Democrats
* Defective and bogus hardware
* Wrenches that dont fit
* Starbucks coffee
* Communism
* Soccer
* The Euro

It's official...

[Your_Mom (94238)]

I can't deny it anymore.

I just read an article on metric pages and found it incredibly intresting.

At one point I said "Wow, Cool"

I think I've gone beyond 'geek'.

I feel dirty.

Re:It's official...

[PaschalNee (451912)]

I feel dirty.
Luckily you can clean yourself up while being compliant to the standard!

In some countries (e.g., Germany) even many brands of toilet paper have format A6.

If you enjoy math..

[teamhasnoi (554944)]

remember that two 8 1/2 x 11 sheets equal an 11 x 17 sheet and four 4 x 5 cards can fit on a 8 1/2 x 11 sheet of paper. A ream of paper is 500 sheets and if you divide that by two, you get 250 sheets which really means nothing; I needed two extra facts for my post about math.

More usefully...

[MartinG (52587)]

you can place two sheets of A4 side-by-side and they will equal an A3 sheet exactly,

More usefully, you can fold an a4 piece of paper in half and it will fit nicely in an a5 envelope.

the metric system is fun

[revery (456516)]

This means that you can place two sheets of A4 side-by-side and they will equal an A3 sheet exactly, and two sheets of A3 will equal an A2."

And of course, 5 sheets of almost any metric sized paper folded into origami lions will inevitably merge to form Voltron, a robot so powerful that it will usually let it's enemies kick it's butt around for a good 15 to 20 minutes before it forms the blazing sword and finishes the fight.

Yes... it's your damn fault!

[Cloud K (125581)]

For those of us living in A4-using lands, it's a real pain in the arse trying to set everything (especially in Windows) from Letter to A4! Then you think you have it susses and sure enough... "PC Load Letter" - aaargh!

Do you have any idea how much trouble and stress you've caused by making Letter the default even with UK set as the country? ;)

metric will win in the end

[garglblaster (459708)]

This is my personal opinion, however:

I think the metric system is like Open Source:
It's going to win in the long run -
simply because it's the logical way to go!

If you look at the evolution of things, there have always been different ways of doing stuff, but in the end one of them won - simply because it was undeniably the best way to go - and the others lost out..

Obligatory homer paraphrasing

[bperkins (12056)]

OOOoo! Look at me! I use the metric system!

I only know how to divide by ten!

Just as Interesting

[Momomoto (118483)]

It's frightening how well they've thought out things like this. From the article:

Technical drawing pens follow the same size-ratio principle. The standard sizes differ by a factor sqrt(2): 2.00 mm, 1.40 mm, 1.00 mm, 0.70 mm, 0.50 mm, 0.35 mm, 0.25 mm, 0.18 mm, 0.13 mm. So after drawing with a 0.35 mm pen on A3 paper and reducing it to A4, you can continue with the 0.25 mm pen. (ISO 9175-1)

Call me an incorrigible geek, but that little tidbit made me giddy.

Not as English as you think....

[rilister (316428)]

Honestly, US companies are genuinely converting to Metric, believe it or not. I work in a consultancy and work with a variety of clients, including a bunch in the worlds of science and medicine.

Since I design things (not code), I have to ask what units they want their things in - I remember one conversation with a wholly US based company going like this:

"What units do you want the database delivered in?"
- [SARCASM BOLD] "We are a scientific company.[/SARCASM BOLD]>
"Oh, right."

They made me feel pretty stupid for asking. I'd say across the product industry it's something like 50/50 right now.

Another Cool Ratio

[donscarletti (569232)]

Although having paper with a 2^0.5 aspect ratio is by far the most practical solution it is not the most aesthetic solution. Photos and slides use an aspect ratio that is not used by any other type of paper. That ratio is (1 + 5^0.5) / 2 or approximately 1.61803399 .

This number is otherwise known as the "golden ratio", it was discovered back in classical Greece and it was known to be the most aesthetically pleasing of all ratios. The Parthenon in Athens was built so that its length and width were dictated by this ratio, it was also used by many Renaissance artists to draw the human body so it seems "perfect".

It is impossible of cause to prove mathematically that this ratio is the best looking of all irrational numbers any more than it is possible to prove mathematically who is the most attractive human, however it's endurance seems to suggest that it has some base to it. It has links with Fibonacci numbers, it also is encountered when drawing regular pentagrams and decagons.

Due to the aesthetically pleasing nature of this ratio I think it would be fairly cool to have a series of paper sizes based on this ratio for artistic uses, rather than the practical but bland "A" series or the fairly pointless American and Canadian series.

Metric & The US

[bobej1977 (580278)]

I used to support using the metric system in the US after going through converting the different units in physics during high school and college, but I've since reconsidered.

I was once talking with some of family and I happened to say something like: 'It was 2 meters from me...' Immediately, one of my uncles interjected a joking comment about how I was the 'product' of the 'new' Math. We then proceeded to go off on a tangent about the merits of the two systems and how expensive it would be to switch to metric.

At that point though, I was struck by how his comment was loaded with negative connotation, which obviously did not stem simply from an aversion to the cost of a hypothetical switch to metric. I realized that the source of his distaste for metric was really just the instinctive reaction social animals use to build communities. The 'Us Vs. Them' filter that we all use to clump ourselves into social groups.

From this perspective, a human perspective, it makes complete sense to have differing systems of measurement. There would be obvious advantages if we all spoke the same language, but no one is proposing that we make everyone learn Chinese (quit being ethnocentric!). Even if everyone DID speak Chinese, people would still use their native languages at home, en familia. Why? Because the stratification of languages helps us to identify our social groups. In this way, we're 'The people who use miles', and they're/you're 'The people who use kilometers'. Communities, when you come down to it, are just sets of these bifurcations.

Taking all that into consideration, I've thrown in with the english system curmudgeons. Why? For the same reason I'm in favor of driver's tests in 16 languages. Because being human ain't about being efficient, it's about communities.

Re:Psst.

[StrawberryFrog (67065)]

Come-on really, Do I want to measure a piece of paper using the square root of two?

No, but it's very pleasant that an A3 page folded in half is exactly the same size as an A4 page. root-two is just the mathematical means to that end.

Re:Oooo.... root 2!

[jdavidb (449077)]

You don't get it. It keeps going, recursively. Two A4's equal an A3, and two A3's equal an A2, and so on. The deal is that the paper is in such proportion that all A* papers are in the exact same proportion. That's not true if you double a 8 1/2 by 11. The proportion there is .77272, while the proportion for a doubled sheet, 11x17 is .647059.

I'm betting the Golden Ratio comes into A4 paper somehow; anyone want to comment?

Re:Oooo.... root 2!

[Pxtl (151020)]

Constant aspect ratio. You don't have to remake your posters/flyers depending on what sheet you intend to print on. 11/17 != 8.5/11. It also makes shipping easier. Basically the only reason to stick with imperial measurements is inertia, as always.

Re:Oooo.... root 2!

[Rupert (28001)]

Those aren't the same shape.

17 / 11 = 1.55
11 / 8.5 = 1.29

A4 paper is twice the size and the same shape as A5. A3 is twice the size and the same shape as A4. It goes up to A0, which is (ooh) 16 times bigger than A4.

Re:Neat

[the MaD HuNGaRIaN (311517)]

did you get a box of 11.69" envelopes to go with that?

Re:Drugs teach American kids the metric system.

[Ubergrendle (531719)]

Origin of the phrase "one for the road". In London, while on the way to the gallows, the cart would stop at each pub along the way. The criminal would be allowed a drink at every pub, almost always 'on the house' so that the soul would not come back to make due on a debt. Also, i suspect pity played a large role.

Re:Drugs teach American kids the metric system.

[Sirch (82595)]

Not quite. The saying actually refers to the trip from the prison to the Tyburn Tree in London. The prisoner to be hanged would be given drink to calm him down for the hanging. The closest pub to the place of hanging that lay upon the route was a mile away. The prisoner would have a drink at this last pub, and then be given a drink to have on his way to the gallows. Interestingly, this is also the origin of "on the wagon" as one of the guards travelling with the prisoner was not allowed to enter the pubs with him. So couldn't drink, and had to stay on the wagon.

Some lovely linkage:here [idler.co.uk], here [phrases.org.uk] and here [looktours.com].

Re:Drugs teach American kids the metric system.

[compro01 (777531)]

if i remember corectly a "hogshead" is 63 gallons.

and a rod is 5.5 yards or 16.5 feets so....

damn your car is a gas guzzeler!

504 gallons to go 1 mile!

Re:Drugs teach American kids the metric system.

[poot_rootbeer (188613)]

504 gallons to go 1 mile!

Yep, sounds like a True American car to me.

Re:Drugs teach American kids the metric system.

[WormholeFiend (674934)]

but metric paper makes much better airplanes.

Re:A complex way to point out simplity.

[Tony-A (29931)]

Gee, I just learned that if you take a sheet of A3 and cut it in half, that's A4.

If you take a D-size sheet of drafting paper,
cut into halves, you have two sheets of C-size drafting paper
cut into quarters, you have four sheets of B-size drafting paper, aka quarto
cut into eight pieces, you have eight sheet of A-size, aka letter, aka octavo.

The metric sizes preserve aspect ratio, the english sizes do not.

Not English

[Gonoff (88518)]

The metric sizes preserve aspect ratio, the english sizes do not

The sizes of paper you use are not English. In England, and the rest of this country, we use the international standard that includes A4. I suspect that you can buy other standards but I have no idea whatI would have to do if I needed "letter" or "legal" size paper.

Re:A complex way to point out simplity.

[ZeLonewolf (197271)]

No, but since you ask:

The number is the measurement below the bust.

an A-cup is a 1-inch difference between the measurement below the bust versus around the bust.
B-cup is 2 inches, C-cup is 3 inches, etc.
DD is the same as E, DDD is the same as EE which is the same as F. This holds valid through an H cup. After that, the interval is 2 inches, with the doubled letter being the in-between value.
This, H-cup is 8", and I-cup is 10", and a 9" difference would be an HH-cup.

The largest bra size manufactured without a special order is a size 60N.

Re:I will fight this metric paper with every OUNCE

[Frothy Walrus (534163)]

Now is that a solid ounce, a fluid ounce, or an apothecary ounce?

Re:Yet another reason for the US to switch to metr

[SkankhodBeeblebrox (581971)]

Your government did try to switch over, not once [wa.gov], not twice, but three [gsa.gov] times! (with limited success, according to the 2nd link)

Also according to that 2nd link,

"Federal agencies were required by this legislation, with certain exceptions, to use the metric system in their procurement, grants and other business-related activities by the end of 1992. "

Not sure what that means to a typical U.S. Citizen, but it appears the U.S. will be metric someday :)

Re:Yet another reason for the US to switch to metr

[madman101 (571954)]

IMHO, the metric system is doomed in the US because it's not American. That's not meant to be funny, sarcastic, or anything other than a simple fact. Well, perhaps it's meant to be a comment on the American psyche...

Re:Yet another reason for the US to switch to metr

[forrestt (267374)]

Actually, it has to do with apple pie. Since there is nothing more American than apple pie, the apple pie recipe is considered sacred. It has been passed down from generation to generation since the start of this glorious nation. Unfortunatly, it has been passed down on the female side of our ancestry, and we men have been telling our women that:

|------| = 10 inches, when in fact
|---------| = 10 inches.

This has caused them to become totally confused with regard to units of measure, and they are thus unable to convert imperial to metric units. Thus, if we were to switch to using the metric system, we would no longer be able to bake apple pies, a situation we are just not willing to accept.

Re:Yet another reason for the US to switch to metr

[gfxguy (98788)]

BTW the metric system sucks for times when you really do want to use fractions

Can you describe a few of these times? I'm being serious... as a novice work worker and DIY home improvement and maintenance guy, I find using mixed fractions very annoying. Yes, you get accustomed to them, but I hardly say that makes it acceptable (hey, people get accustomed to Windows crashing, and find it acceptable to have to reboot or reinstall - I'm not one of them).

Besides, it's not like you can't use fraction in metrics, either - so you say 1/2 cm instead of 5 mm, if it floats your boat.

Re:Huh?

[rnelsonee (98732)]

I can... metric is obviously heavily biased to factors of 10 - a pretty useless number as it only has a few divisors (2, which always a factor of the base/radix used, and 5). So when you want to divide your 1 meter board into 4, you're now stuck with 2.5 cm pieces. Okay, not too bad. What about dividing it by 3? A repeating decimal is now your new measurment. Bah, same with 6. If you get stuck in classes for most of your life, you probably won't care about this. But try to build a bar with a metric ruler and you'll see why these are desirable traits.

Compare that with, say, the foot. 12 inches - easily divisible by 3, 4, and 6. Makes building that shit a lot easier :)

Same goes with volumes - it's easy to convert gallons to quarts to pints. You have to memorize more units (which I agree sucks), but it makes making that recepie easier when you realize you have more guests coming.

Re:Yet another reason for the US to switch to metr

[Too Much Noise (755847)]

What if you have to divide by 5? or 8? How heavy is each patty then?

It's easy to pick the numbers you like. There are always numbers a given multiplier won't divide nicely to.

Re:Yet another reason for the US to switch to metr

[fireduck (197000)]

except that the US apparently sanctioned the metric system in 1886, and the American Bureau of Standards made the metric system it's standard in 1964. (nice timeline here [miketodd.net]). There've been various attempts to further adopt in more recent history, but basically the US doesn't want to change. The metric system is nonexistant as far as general use is concerned. The only "off the top of my head" metric use I can think of are 2 L bottles of coke. nothing else gets metric treatment.

Re:And this is superior why?

[Moderation abuser (184013)]

Because you can scale an A4 page down to A5 size and print two of them on each A4 sheet and they fit *exactly*. Saves paper.

If you are a *real* skinflint and have good eyes you can scale down to A6 and print double sided. It works quite well with a decent laser printer.

The reverse is true obviously if you want to scale up. You can tape (A4 usually because it's the most common) pages together to make A3, A2 and A1 sheets and it all fits together exactly.

Having said that, I kind of assumed that the same thing applied to US paper sizes. Surprised it doesn't.