Ever downloaded a document or received a file from a friend only to have it print out badly? There are lots of potential causes of such problems: different typefaces available to the creator and the printer; different operating systems and different versions of the same operating systems exposing limitations in supposedly cross-platform standards; different printer engines, especially when crossing between ink-jet and laser; and so on.
An oft-overlooked cause of problems, however, is different paper sizes. When people in the US and Canada reach for a sheet of paper to write or print on, chances are they reach for a piece of Letter-sized paper (also known as US Letter), measuring 8.5˝ by 11˝. With few exceptions, when people everywhere else reach for a sheet of paper to write or print on, they reach for a piece of A4-sized paper, measuring 210mm by 297mm.
A quick conversion between inches and millimetres shows the two sizes aren’t all that different:
| | Millimetres | Inches | |
 |
| | Width | Length | Width | Length |
| |
| A4 | 210.0 | 297.0 | 8.26 | 11.69 |
| Letter | 215.9 | 279.4 | 8.50 | 11.00 |
| |
And a scale representation of each page size reinforces the closeness of the two paper sizes.
Which raises the question, why the difference at all? If both sizes are arbitrary, why bother with maintaining a difference. The answers are long and involved, and mostly outside the scope of this article. At the core, however, it comes down to one thing: A4 isn’t an arbitrary size.
A4 Described
A4 is part of the ISO 216-series of related paper sizes known more commonly as the A-series. This series starts with the large A0 sheet and a quick look at this large sheet of paper shows why these various sheets are the sizes they are.
A0 sheets of paper are 841mm by 1189mm. Again, apparently arbitrary. Multiply the two numbers together, however, and it becomes a little clearer: 841 * 1189 = 999,949mm squared or 0.999949m squared (ie just a smidgen under a square metre of paper). For all practical purposes, an A0 sheet contains a square metre of paper.
So why not make it a 1m by 1m sheet? Because of another non-arbitrary consideration: the aspect ratio or relationship between the height and width of each sheet.
1189/841 = 1.413793103448276, not particularly memorable, unless you happen to be maths-geeky enough to see the similarity between it and √2 (the square-root of 2, an irrational number which starts thus: 1.414213562373095) Round both numbers to four significant figures and you get the same value: 1.414.
So, the aspect ratio of an A0 sheet of paper is, again for practical purposes, one as to the square-root of two or 1:√2. And again, I hear the cries: ‘so what!’
A ratio of 1:√2 is more than a mathematical oddity. It doesn’t have a nifty name, like the famous Golden Ratio or Golden Mean. It does, however, have a nifty property. Divide a rectangle with sides 1:√2 along the longest side and the smaller rectangle you create has the same aspect ratio. (Markus Kuhn suggested in correspondence we call the ratio the Lichtenberg Ratio, after Professor Georg Christoph Lichtenberg, the German enlightenment figure who first proposed the ratio as a basis for paper formats in 1786.)
Getting back to the ratio (named or not) and its nifty property: if we start with a honking great sheet of A0 paper:
We can easily, and quickly, derive all the other A-series sizes by folding or dividing thus:
In less visual terms, any sheet of A-series paper is as long as the next-larger sheet is wide and half as wide as the next-larger sheet is long. To wit:
| Sheet name | Width (mm) | Length (mm) | |
 |
| A0 | 841 | 1189 |
| |
| A1 | 594 | 841 |
| |
| A2 | 420 | 594 |
| |
| A3 | 297 | 420 |
| |
| A4 | 210 | 297 |
| |
| A5 | 148 | 210 |
| |
| A6 | 105 | 148 |
| |
| A7 | 74 | 105 |
| |
| A8 | 52 | 75 |
| |
There are other benefits to this relationship between paper sizes, not least of which is when you want to scale a particular layout. If you’ve ever wondered why photocopiers offer a 71% reduction option wonder no more: 0.71 is approximately equal to (√2)/2 or √0.5. This makes it perfect for reducing an A3-layout onto an A4 sheet, or an A4 layout onto an A5 sheet or, more commonly, reducing two A4-sheets side-by-side — say in a journal — neatly and without fuss onto one A4-sheet. The equally common 141% option is, of course, perfect for enlarging from one A-series sheet up to the next (eg A4 to A3). Most important, because each sheet has the same aspect ratio, objects retain their relative shapes: squares don’t become rectangles and circles don’t become ellipses.
If nothing else, this constancy of relationship makes A-series paper simpler to work with than older paper sizes such as Brief (13˝ by 16˝, and the source of the ‘briefs’ lawyers still use) or Foolscap (27˝ x 17˝) and its near-letter sized derivative, Foolscap Quarto (13.5˝ by 8.5˝, commonly if erroneously called ‘Foolscap’).
Add in a clear connection to the metric (or, more properly, the SI) measuring system and the rise in popularity of A-series paper is fairly easy to understand: as the world has slowly but surely gone metric, so A-series paper has become more popular. In Australia, for example, the metric system was adopted officially in 1974, the same year A-series paper (and related series such as the C-series for envelopes) started to become the standard.
US Letter Described
The clear connection to the metric system is also a partial explanation for the continued use of Letter-sized paper in the US and Canada. The US is almost the only country left not to have made the switch from non-metric measures, making the particular advantages of A4 less evident. As well, although US paper sizes are as arbitrary as is sometimes contended, they aren’t impossible to work with.
There is no derived starting point (equivalent to the 1 square metre for A0) for US paper sizes but the two most popular sizes — Letter and Tabloid — are part of an old American National Standard Institute standard for technical drawing paper. This standard (ANSI/ASME Y14.1) had five paper sizes swinging back-and-forth between two different aspect ratios:
| Sheet name | Width (˝) | Length (˝) | Aspect Ratio | |
|
| A (Letter) | 8.5 | 11.0 | 1.294 |
| |
| B (Tabloid) | 11.0 | 17.0 | 1.545 |
| |
| C | 17.0 | 22.0 | 1.294 |
| |
| D | 22.0 | 34.0 | 1.545 |
| |
| E | 34.0 | 44.0 | 1.294 |
| |
This isn’t as elegant or convenient as A-series paper but enlarging and reducing particular layouts whilst retaining internal relationships isn’t especially difficult. Just skip a paper size when travelling in either direction.
It’s worth noting that neither aspect ratio has any particular mathematical properties. And there being two aspect ratios isn’t surprising: fold any rectangle in half that doesn’t have sides in ratio 1:√2 and the smaller rectangle’s sides will be in a different ratio to each other. Fold the smaller rectangle in half again and this third rectangle will have sides in the same ratio as the one you started with.
This simple property is why rectangles with sides in ratio 1:√2 are so nifty: they are the only ones in which the two ratios you get folding back-and-forth are equivalent and interchangable.
And the sheer utility of this interchangability is why I believe older paper sizes such as US Letter will eventually disappear, even in the US. For example, the current version of the ANSI standard noted above — ANSI/ASME Y14.1m-1995 — recognises the older paper sizes for legacy purposes only, setting A-series paper as the preferred US standard for technical drawings.
Moreover, I understand A-series paper — especially A4 — is slowly becoming the norm in US colleges and universities, if for no other reason than making it easier for students and staff to photocopy articles from (inevitably A4-sized) journals.
Finally, globalisation exacts its toll: US companies doing business with officialdom outside the US (especially the EU) are finding they must submit proposals, tenders, diagrams and so on on A-series paper.
[From Between the Borders]