How is the Size of A4 Paper Determined?

The table below shows the ISO A series paper sizes (in mm) from A0 through A10. The most common one is A4, of which the exact size is 210 × 297 mm. How is that size determined?

Amazingly, you can exactly compute all these sizes from the following two simple rules.

1. The area of a sheet of A0 is 1 m², with the actual paper size rounded to the nearest mm.
2. When you fold a page in two, the size of the folded page has to be equal to the next smaller size (for example, a folded A3 must be the size of an A4), with the actual paper size rounded down to the nearest mm.

Applying the Rules

Rule 1 is simple, you have to start from some basic size, and 1 m² is a nicely round number.

Rule 2 exactly determines the aspect ratio of a sheet, since there is only one aspect ratio for which it holds. Why is that so? If you fold a sheet of size \(w\times h\), you get a sheet of size \(h/2\times w\). If you want the aspect ratio of both sheets to be the same, you need to have

\[\frac{w}{h}=\frac{h/2}{w},\]

which means that \(h/w=\sqrt{2}\). Hence, the ratio between the width and the height of a page has to be \(1/\sqrt{2}\). Rule 2 allows for easy scaling. For example, a copier can shrink an A3 page onto A4 paper without fiddling with margins.

With the area of a sheet of A0 and its aspect ratio known, we can now compute its exact size. We have that \(hw=10^6\,\mathrm{mm}\) and \(h=\sqrt{2}w\). It follows that \(\sqrt{2}w^2=10^6\), from which we get that \(w=841\,\mathrm{mm}\) (after rounding to the nearest mm). From \(hw=10^6\,\mathrm{mm}\) and \(w=h/\sqrt{2}\), it follows that \(h=1189\,\mathrm{mm}\) (again after rounding to the nearest mm). This makes the size of a sheet of A0 841 × 1189 mm.

The dimensions of the smaller sizes are then computed one after the other by dividing the largest dimension by two (and rounding down). For A1, we get that \(\left\lfloor{1189/2}\right\rfloor=594\), so A1 is 594 × 841 mm. Continuing in this way, we arrive at 210 × 297 mm for A4. The following Python script computes sizes A0 through A10.

from __future__ import print_function
from __future__ import division
 
width = int(1000 / 2 ** (1 / 4) + 0.5)
height = int(1000 * 2 ** (1 / 4) + 0.5)
for i in range(11):
    print('A' + str(i), '=', width, 'x', height, 'mm')
    width, height = height, width
    width //= 2

Some Other Interesting Tidbits

The weight of paper is typically expressed in g/m², which immediately provides us with the weight of a sheet of A0. For a typical weight of 80 g/m², we get exactly 5 g for a sheet of A4, since its area is that of A0 halved four times, so divided by 16.

When following the rules through for even smaller sizes than A10, we arrive at the smallest possible size being A19. Curiously enough, A19 is actually a square of 1 × 1 mm. I hope nobody is crazy enough as to actually produce A19 paper. The next smaller size, A20, would then be one-dimensional (0 × 1 mm), which is not physically realizable, and all following smaller sizes are 0 × 0 mm. But I’ll stop there, since we have clearly descended into the realm of silliness.