# Reason for A paper size width/height ratio

Dear Straight Dope,

“All A sizes are in the proportion 1 wide by the square root of 2 deep (1:1.414…). It’s a bit compulsive and you will not be surprised to learn it was thought up by the Germans.”

Actually there’s a good reason for it to be like that. Cecil even mentioned the effect:

“You fold [A0 paper] in half to get A1, you fold THAT in half to get A2, till eventually you get down to A4, A5, A6, etc.”

And that’s only possible if the width/height ratio is 1/sqrt(2), so there’s nothing compulsive about that, it’s just sensible to make it that way.

Yours, Ermel.

One person’s compulsive is another person’s sensible. You can always fold and refold paper of any ratio, it’s just that the ratio won’t stay the same except for the sqrt(2) one. So the question is: Who really cares that the ratio of various paper sizes have to stay the same? Insisting that they all have the same ratio might be viewed by some as compulsive.

Note that sqrt(2) is easy to measure using classic geometry, so that might have something to do with the choice as well.

All the people who work with paper on a regular basis in an office/ filing area care, because if it’s the same ratio, it makes filing and mailing and so on much easier.

It also makes it possible to print the same document on larger or smaller stock (or print multiple pages per sheet) without changing the layout or proportions of individual pages at all.

Constanze: That’s an argument for standard sizes, not ratios.

Which is almost a good point in the computer age, but when the paper sizes were decided, if you wanted to rescale a document, you had to start from scratch. (And given the limits of hand-set metal typefaces, you probably couldn’t use the old version as a layout guide anyway.)

But… it has been my experience in scaling documents to different paper sizes, keeping the margins looking right makes a mess of things. And when scaling up, you have to go with 2 column formats which changes things a lot.

I’ve published a lot of scientific papers over the years. The first version is a TR on standard 8.5x11. The next is a conference version, slightly bigger paper, two column (but not a bigger font). The last version is the journal version, usually a fairly small page size. None of them look at all alike. What fits on a line (code, equations, etc.) changes quite a bit.

Good point about the scaling of documents, Mangetout.

Another thing that’s easier with a 1/sqrt(2) ratio is printing brochures and booklets. Whatever method you use to assemble them, you’ll just need some A paper size to do it.

It’s also easy to make some A5 paper when all you have is A4. You don’t even need a measure.

ftg: Your last paragraph sums it up nicely. No such triple workload for us. Just use A4 and be done with it. I used to edit a club magazine: typeset and proofed on A4, than scaled to A5 for printing. Worked like a charm.

Not to mention that it makes it easier to adjust the output run of a paper factory to meet market conditions.

It’s nothing to do with rescaling documents, it’s a simple matter of paper size in manufacturing.

You make size A, a broad sheet of some sort. You need more manageable sheets, cut it in half, half again, etc. Half is easier than thirds or fifths. Each sheet is approximately the same general shape. Once there’s a standard size, more or less, that’s how big they make drawers and filing cabinets.

The real question is how we ended up with letter and legal sizes? What, lawyers talked and wrote too much? (Couldn’t shut up?) Or they double-spaced everything and needed more paper? Why is it called “foolscap”?

Huh? Only if you never had to work with A paper size, I assume. Because it certainly helps that folding one A3 in half, I can file it with the rest of the A4s, or put two A5s side-to-side in one A4 sleeve … that does makes things easier than changing ratios.

I didn’t realize so many Dopers … oh well.

Let’s do a simple example. Suppose you had two paper sizes Z3 and Z4. Z3 is 10x8 and Z4 is 8x5. Note that one can cut a Z3 into 2 Z4s quite easily. You can file two Z4s side-by-side in a Z3 folder or fold a Z3 and put it into a Z4 folder. Properties that have nothing to do with a sqrt(2) ratio.

In other words: The ratio doesn’t matter at all for most of the arguments people are making. The scaling, as I mentioned and which is almost workable, is not as great an advantage as some people think it is. And the origins are pre-computers.

If you wish to argue the advantages of a sqrt(2) ratio, maybe you should think first if it also applies to other ratios. And in particular, if preserving the ratio across sizes must be required.

I don’t understand any of this and I’m intrigued. Are you sure you are referring to ‘scaling up’ and not some other process(es). If the margins are aesthetically correct for an A4 version of a document, they should remain so whether you scale it up or down to any other A size paper. And scaling up has nothing to do with the number of columns you would use to present the same information.

I’m sorry they don’t have Wikipedia where you live.

Scaling may not be that big an advantage, but it is an advantage, so, all else being equal, it makes sense to prefer it. Standards are arbitrary anyways, so even a slight advantage can help one win out over another.

And, while scaling before computers would often require the typesetter resetting everything, at least he would know exactly how to lay it all out. He doesn’t have to sit there figuring out again how many words should fit on a particular line, for example.

Also, by reading the official history, it does appear that the concept was done by one man for more of a pure mathematical sense–A0 was designed to have an area of exactly one square meter, because basing it purely on lengths might mean the person would measure incorrectly. Having OCD myself, I can see a bit obsessiveness in this man.

The whole history of paper use is based on simply folding sheets in half to make smaller versions. That works no matter what the ratios are.

Newspapers, larger then than today, were folded in half to print tabloids. The earliest comic books were half-tab or tabloid paper folded in half.

Ever hear of folio in books? That was the name of the largest sized book paper, although the actual size varied in practice. Whatever the number of inches, a folio sheet printed four pages. Then came quarto, 8 pages, octavo, 16 pages, sixteenmo, 32 pages, thirty-twomo, 64 pages, and sixty-fourmo, 128 pages.

Letter size paper 8 1/2 x 11 scales up to ledger at 11 x 17 and then up to C, D, and E or 17 x 22, 22 x 34, and 34 x 44.

Many trade paperbacks today are printed at half letter size, 5 1/5 x 8 1/5 instead of ALA octavo, which is 6 x 9. Half letter size is easier to work out on a computer and home printer since you can do fit four pages properly onto a page out of a ream of copier paper.

I can’t understand what you mean by this. Of course you have to refigure the number of words on a given line. In the days of setting lead type, you couldn’t scale it down that way. Neither typography nor readability works that way. The type size for a trade paperback is not half that of a large hardback. It likely is the same. You can’t use the same margins or leading for a book page twice as large or half as large either. Everything needs to be redone from scratch.

Not only folding, but also cutting, with a minimum of waste.

Intermediate paper suppliers get large sheets (or rolls), cut and repackage into smaller sheets. The less wasted, the more profit.

The point isn’t that A3 is twice as big as A4. The point is that A3 is twice as big as A4 and is also the same shape. Your Z3 paper is twice as big as Z4, but they aren’t the same shape. However, Z5 (made by cutting Z4 in half) is the same shape as Z3, but with both linear dimensions 1/2 as big.

This works because x:1::1:(x/2) only when x is the square root of 2.

I believe that the important advantage, as has been mentioned, is not scaling documents but rather scaling paper. Using A size papers, the manufacturer only needs to make one size of paper, A paper. All the other sizes can be created from a sheet of A paper by simply cutting it in half.

If we used your Z scale of paper, then a sheet of Z paper would be 20x32. When you cut the paper in half repeatedly, you get sheets which are alternately shaped like 5x4 and 8x5. These sheets are not the same shape, at all.

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(note that no allowance has been made for the fact that an ascii character is not a square. )

Just to make it 100% absolutely clear. I am fully aware of the special property of sqrt(2) insofar as preserving the ratio.

Got that? Anyone not able to understand this?

In particular, Tenebras, just FYI. I have PhD in Computer Science. I can do basic Math. I am aware that 5x4 and 8x5 are not the same proportions. In fact that was the whole point of the example! (I would really like an explanation as to why you would suspect I wasn’t aware of this.)

That in no way explains why paper makers have to choose that ratio. It seems to me to not be so much sensible as compulsive. (Remember the OP?)

There is a great deal on the subject in this article. Note that the existence of sqrt(2)-aspect paper goes back at least to the mid 18th century, long before the ISO standards or the preceding DIN.

Note also that the full standard is not limited to the A series. There is also the B series, each one midway between two As, and the C series, midway between A and B, which is used for envelopes for the A series. (For example, a sheet of A4 goes flat into a C4 envelope, and, folded once, into a C5 envelope.)

I was picturing remaking something into a large print version, like my Grandpa’s Bible. To me, your examples would not actually be “scaling”, even if we were talking about doing so on computers. Scaling implies that you kept all the proportions the same, including the text size and margins.

I do want to thank you for being nice, and just asking what I meant, rather than being rude about it.

Scaling or not, no large print version was ever created like this. It’s not the way typography works.

You must have caught me on an off day.