Tag Archives: the importance of the square root of 2 in paper sizing

A Measure Of Things – Part Eleven

Having explored the unnecessary complications of the British Imperial paper size system, let’s get up to date. Paper, at least in Europe and the UK – we will deal with the pesky Americans later – comes in what is known as the A-series of sizes. Office workers will be most familiar with the A3, the A4 and the A5 sizes, although the starting point is the A0. The A0 paper size is exactly I square metre in area, although its dimensions are 841 millimetres by 1,189 or, if you prefer, 33.1 inches by 46.8.

What underpins the A-series of paper sizes is that the height to width ratio of all the sizes is the square root of 2 or 1.4142 to 1, if you prefer. It was a German scientist and philosopher, Georg Christoph Lichtenberg, who first noticed in 1798 how useful exploiting a standard height to width ratio would be in dealing with paper. If you cut a piece of paper parallel to its shorter side to make two equal pieces, each of the resulting pieces would have the same height/width ratio of the square root of 2.

So the A1 size is half that of the A0 and is derived by halving the larger piece across its larger side. A2 is half the size of the A1 and so it continues – the A3 being half the A2, the A4 half of the A3 and the A5 half of the A4. You get the drift now with each size retaining that all important common height to width ratio. Clever, really. It was not until the 20th century that Lichtenberg’s observations were put into practical use, Dr Walter Portsmann creating a defined system of paper sizes, which were adopted in Germany in 1922 as the DIN standard.

The overpowering logic and convenience of the system meant that it was rapidly adopted elsewhere – even by the Brits in 1959 – and it became the internationally recognised standard by 1975. But what about B-series paper and C-series envelopes, I hear you cry. Well, not all of the many paper formats conform to the A-series and so the B deals with them but they are linked to the A-series. The B1 size is the geometric mean between the A0 and the A1 and so on and, of course, retaining the all-important height to width ratio of the square root of two. The C-series relates to envelopes and it is based on the geometric mean between the A and B-series of the same number. So that is why if you get a C4 envelope an unfolded A4 piece of paper fits in it like a glove.

Adopting a more metric-based system of paper sizing encouraged the Brits to change the quantities in a quire and ream to 25 and 500 respectively which is why you now buy your computer paper in bundles of 500.

In the USA, Canada, Mexico and a few other countries, the ISO 216 standard has not been adopted and they use letter paper which at 8.5 inches by 11 is slightly wider and shorter than A4, legal which is 8.5 inches by 14 and Ledger or Tabloid which is 11 inches by 17. The letter size only became a recognised standard in the States in 1921, although, somewhat bizarrely, the US government didn’t adopt it until the early 1980s, adopting a size of 8 inches by 10.5 in the interim. Presumably, the increasing sharing of digital documents forced their hand. The major problem with the American system, aside from the inconvenience of sharing and printing documents between countries adopting the different standards, is that it does not have the standard height/width ratio and so switching from one size of paper to another can cause no end of formatting problems.

It will only be a question of time before they adopt the A-series, methinks.