Why A4? – The Mathematical Beauty of Paper Size

Ben Sparks

Unless you have skipped a lot of school, or work, or both – or you live in the USA – you have probably used an A4 sheet of paper before now. Have you ever wondered why it is the shape or size it is? Time to dust off some high-school level maths to investigate.

Just in case you have never actually checked the size, I can tell you now it is precisely 210mm by 297mm. But don’t take my word for it. Go check. I’ll wait. If you haven’t got a piece of A4 within reach somewhere in your house then you have a much better tidying ethic than me. Conveniently, it is just short enough to be measured by that old 30cm ruler you also probably have lying nearby in a drawer somewhere. Nevertheless, the obvious question remains:

Why?

Sometimes this is closely followed by: “Seriously? 297mm? Why not 300mm?”. I know this is a genuine question because the words still echo in my memory from my school design classroom where one of my fellow students railed at the perversity of the world in that exact manner. My memory does not supply any detail about whether he received an answer, but I fear he was left to wallow alone in his own misery. Let us consider the rest of this piece as a response to this poor boy’s unresolved anguish.

It is true though – anyone who has ever wanted to mark half way across the long side of their paper has experienced the vague resentment that occurs when you realise you now need to measure 148.5mm and your ruler does not even have half millimetre divisions.

I invite you to take a rectangular piece of paper that is not A4 sized. You can always just tear a bit off your A4 paper and then neaten it up to a rectangle. With your non-A4 rectangle, try folding it in half along the shortest line of symmetry. You will observe, in a spectacular anticlimax, that you now have a piece of paper half the size, and a different shape. Possibly, you started with a ‘squarey’ rectangle and now you have a ‘long-thin-rectangle’, or vice versa.

Now do it with an A4 sheet. You probably already know what happens. You get an A5 piece of paper. It is half the size (of course it is, you just folded in half). What’s more, it is the same shape. Technically a similar shape, of course, but the sides are in the same ratio. This is something of a shock, if you ponder it, because rectangles do not normally behave like this.

This is not an accident. It is possibly one of the greatest innovations of the 18th century. To take just one modern example: teachers have been using it to literally halve their photocopying budget for years. You want two copies on one page? Great – they fit exactly! Any other paper shape (say, ‘letter size’, or 8.5 by 11 inches, for all you North Americans out there) is sadly wasteful in comparison because your two half size copies leave an awkward gap on the original page.

The earliest recorded discussion of the idea comes from a 1786 letter from German academic Georg Christoph Lichtenberg to author Johann Beckmann1, but there is a suggestion it might have already been a problem used in a maths exam even earlier.2 However, it was not until the early 20th century that Germany – and then eventually most of the rest of the world – actually standardised the idea. It is now known as ISO 216, an international standard paper size.3

In fact, there is only one ratio of rectangle sides that will work, i.e., one that will give a similar shape when cut in half. Consider for yourself the question: Which ratio is it?4 Read on to see one possible method of settling this.

Draw a general rectangle, with sides in the ratio (long:short) \( x:1 \)

Rectangular paper with sides in ratio 1 to x

Now draw the half-way fold line, and consider the new rectangle which has a side ratio (long:short) of \(1:\frac{x}{2}\)

Rectangular paper split in half so that the new smaller rectangle has sides in ratio x/2 to 1

If we wish the two ratios to be the same for all the aforementioned goodness then we must have the fractions equal

\( \begin{array}& \frac{x}{1} &= & \frac{1}{\frac{x}{2}} \\ x^2 &= &2 \\ x &= &\sqrt{2} \end{array} \)

(or \( – \sqrt{2}\), but let’s discuss negative ratios another day)

So the only ratio that has this important property is the square root of 2, famously – and ironically in this case – not a ratio5. This is why the dimensions of the paper, whatever unit you use, are never going to be ones which most people would call ‘nice’. There just don’t exist a pair of integers that give you the ratio \(\sqrt{2}\), so we have to approximate.

We should (and did) therefore abandon the dream of having ‘nice’ side lengths, but that does not stop us from making ‘nice’ areas. In fact, the international A-size system now starts with A0 paper, with sides in the correct \(\sqrt{2}\) ratio, but with exactly 1m2 of area, or as near as you can get with whole millimetre side lengths (1189mm  841mm, to be precise). Then folding or cutting it a few times until it is conveniently desk/folder sized gets you to A4, hence the ‘4’ bit in the name.

By the way, if this is new to you but you do remember a bit of school maths, you might like another way of realising this: remember ‘area scale factors’ and ‘length scale factors’ are related in that the area scale factor is precisely the square of the length scale factor. Well, if you want the area scale factor to be \(2\) (or \(\frac{1}{2}\) ) then the length scale factor must be \(\sqrt{2}\) (or \(\frac{1}{\sqrt{2}}\) ). Of course.

It is somewhat sobering. Irrational numbers are useful, whether you like them or not (a lot of people do not, just check some of the myths that have grown up around what the Pythagoreans did to Hippasus when he suggested \(\sqrt{2}\) was not rational). But once we get over our fear of using these irrationals we can reap the benefits. In this case, paper weights are now easily calculated because weight is proportional to area: 80 gsm (grams per square metre) paper at A0 size weighs precisely 80g. A4 paper at that density therefore weighs 5g since it’s been halved 4 times.

Draughtsperson’s pens typically have widths that increase by factors of \(\sqrt{2}\approx 1.4\) so that the next pen in the series can have the correct width if used on a scaled up drawing on the next size of paper. It’s all nice and neat.

Some technical pens with their sizes marked (increasing by factor of approx 1.4 each time)

Some technical pens with their sizes marked (increasing by factor of approx 1.4 each time). Image Credits: Wikimedia Commons

What is our lesson from all this? We often have convenience now because someone did the maths before. Thanks to them, we can now safely forget that even some of the most basic aspects of our lives are governed by numbers and their properties. But they are, and we benefit greatly from the people that have reflected upon this.

I say we can safely forget all this, but I will end with a request: the next time you hear the cry “But WHY is it 297mm?” from a nearby anguished soul trying to measure halfway across their paper, bring it back to mind. Now you know.

 

 

 


[1] Lichtenberg is the same Lichtenberg who gives his name to the striking lightning burn patterns from high voltage incidents. Beckmann, for his part, was the very first person to coin the word ‘technology’. Now you know.

[2]https://www.cl.cam.ac.uk/~mgk25/lichtenberg-letter.html

[3]https://en.wikipedia.org/wiki/Paper_size

[4] No. It is not the Golden Ratio. That is a great ratio, but it is not this ratio.

[5] Not a ratio of integers. Look up a proof if you don’t believe me.

The post Why A4? – The Mathematical Beauty of Paper Size originally appeared on the HLFF SciLogs blog.