# Why is a piece of paper the size it is?

There is an international size of paper which has an aspect ratio of the square root of two.

At which point you might not be any wiser than when you started reading this, so let me try to explain.

The aspect ratio of paper expresses the length of one side of the paper against the length of the other side.  So if you had a piece of paper 200mm top to bottom and 100mm left to right, you would have an aspect ratio of 200:100 or more normally written, 1:2.

For reasons that I have never quite worked out we don’t have paper on a ratio of 1:2 but rather on a ratio of 1:1.4142.  1.4142 as you may have guessed by now is the square root of two.

The base size of paper based on this aspect ratio is called A0 size, and the special thing about it is that it has an area of one square meter.  Using the aspect ratio of the square root of two, you end up with a piece of paper on which the sides are 841mm and 1,189mm (33.1 × 46.8 in).

To move down the size of paper through A1, A2, A3 and then ultimately A4 (which we are all familiar with) you keep folding the paper in half along the larger dimension.  So a sheet of A3 is effectively two sheets of A4 side by side. Fold A4 in half and you get A5 size.

Now mathematicians get very excited by this because it turns out that when you fold a sheet of anything paper in half the aspect ratio stays the same, so all A paper (A1, A2, etc, etc.) has an aspect ratio of the square root of 2. Isn’t that exciting!

OK, I was being sarcastic in writing “isn’t that exciting” but there is a point here. Because the ratio always stays the same you can change the size of the paper without losing anything off the edges when you use a photocopier to enlarge or reduce without worrying about the fit.

There’s another benefit of the A system and that is that it makes weight easy to calculate. A sheet of A4 made from 80 gram/m² paper weighs 5g.  This turns out to be a very helpful fact since 80gsm paper is the most common paper in the known universe. Since we know that postage rates change at 100g it is easy to see that 20 sheets of A4 weigh 100g.

2A0 (at 1,189 x 1,682 mm or 46.8 x 66.2 in) is just about the largest piece of paper measured by this system which is normally available, and at the other end A7 is the smallest, being somewhere around half the size of normal postcards (A6).  A7 is 74 x 105 mm or 2.9 x 4.1 in.

Of course A size paper won’t fit in A size envelopes, so there is also the C series, which works in the same way with the same ratio, but starts from the premise that a sheet of A4 paper can fit into a C4 envelope without folding. In between the two is the B series, in which the shorter side of B0 is exactly 1m. There is even a D series, although as far as I know this is only used in Sweden – and I am not quite sure what they do with it.

Fortunately it doesn’t matter what sort of paper you have, we can store it, and that is why Admiral exists.