A4 Paper Stories
By Susam Pal on 06 Jan 2026
I have a measuring technique that is neither fast, nor accurate, nor recommended by any standards body and yet it hasn't failed me whenever I have had to use it. I'll share it here but don't ever use it for installing kitchen cabinets or anything that will stare back at you every day for the next ten years. It involves one tool: a sheet of A4 paper.
Like most sensible people with a reasonable sense of priorities, I do not carry a ruler with me wherever I go. Nevertheless, I often myself needing to measure something at short notice, usually in situations where a certain amount of inaccuracy is entirely forgivable. When I cannot fetch a ruler easily, I rely on the next best thing: a sheet of A4 paper, which is available in abundant supply where I live.
From photocopying night-sky charts to serving as a scratch pad for working through mathematical proofs, A4 paper has been a trusted companion since my childhood days. I use it often. If I am carrying a bag, there is almost always some A4 paper inside: perhaps a printed research paper or a mathematical problem I have worked on recently and need to chew on a bit more during my next train ride.
Dimensions
The dimensions of A4 paper are the solution to a simple, elegant problem. Imagine designing a sheet of paper such that, when you cut it in half parallel to its shorter side, both halves have exactly the same aspect ratio as the original. In other words, if the shorter side has length \( x \) and the longer side has length \( y , \) then \[ \frac{y}{x} = \frac{x}{y / 2} \] which gives us \[ \frac{y}{x} = \sqrt{2}. \] Test it out. Suppose we have \( y/x = \sqrt{2}. \) We cut the paper in half parallel to the shorter side to get two halves, each with shorter side \( x' = y / 2 = x \sqrt{2} / 2 = x / \sqrt{2} \) and longer side \( y' = x. \) Then indeed \[ \frac{y'}{x'} = \frac{x}{x / \sqrt{2}} = \sqrt{2}. \] In fact, we can keep cutting the halves like this and we'll keep getting even smaller sheets with the aspect ratio \( \sqrt{2} \) intact. To summarise, when a sheet of paper has the aspect ratio \( \sqrt{2}, \) bisecting it parallel to the shorter side leaves us with two halves that preserve the aspect ratio. A4 paper has this property.
But what are the exact dimensions of A4 and why is it called A4? What does 4 mean here? Like most good answers, this one too begins by considering the numbers \( 0 \) and \( 1. \) Let me elaborate.
Let us say we want to make a sheet of paper that is \( 1 \, \mathrm{m}^2 \) in area and has the aspect-ratio-preserving property that we just discussed. What should its dimensions be? We want \[ xy = 1 \, \mathrm{m}^2 \] subject to the condition \[ \frac{y}{x} = \sqrt{2}. \] Solving these two equations gives us \[ x^2 = \frac{1}{\sqrt{2}} \, \mathrm{m}^2 \] from which we obtain \[ x = \frac{1}{\sqrt[4]{2}} \, \mathrm{m}, \quad y = \sqrt[4]{2} \, \mathrm{m}. \] Up to three decimal places, this amounts to \[ x = 0.841 \, \mathrm{m}, \quad y = 1.189 \, \mathrm{m}. \] These are the dimensions of A0 paper. It is quite large to scribble mathematical solutions on, unless your goal is to make a spectacle of yourself and cause your friends and family to reassess your sanity. So we need something smaller that allows us to work in peace, without inviting commentary or concerns from passersby. We take the A0 paper of size \[ 84.1 \, \mathrm{cm} \times 118.9 \, \mathrm{cm} \] and bisect it to get A1 paper of size \[ 59.4 \, \mathrm{cm} \times 84.1 \, \mathrm{cm}. \] Then we bisect it again to get A2 paper with dimensions \[ 42.0 \, \mathrm{cm} \times 59.4 \, \mathrm{cm}. \] And once again to get A3 paper with dimensions \[ 42.0 \, \mathrm{cm} \times 29.7 \, \mathrm{cm}. \] And then once again to get A4 paper with dimensions \[ 21.0 \, \mathrm{cm} \times 29.7 \, \mathrm{cm}. \] There we have it. The dimensions of A4. These numbers are etched in my memory like the multiplication table of \( 1. \) We can keep going further to get A5, A6, etc. We could, in theory, go all the way up to A\( \infty. \) Hold on, I think I hear someone heckle. What's that? Oh, we can't go all the way to A\( \infty? \) Something about atoms, was it? Hmm. Security! Where's security? Ah yes, thank you, sir. Please show this gentleman out, would you?
Sorry for the interruption, ladies and gentlemen. Phew! That fellow! Atoms? Honestly. We, the mathematically inclined, are not particularly concerned with such trivial limitations. We drink our tea from doughnuts. We are not going to let the size of atoms dictate matters, now are we?
... continue reading