A4 Paper Stories – Susam Pal

By Susam Pal on 06 Jan 2026

I sometimes resort to a rather common 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 will
describe it here, though calling it a technique might be overselling
it. Please do not 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
find myself needing to measure something at short notice, usually in
situations where a certain amount of inaccuracy is entirely
forgivable. When I cannot easily fetch a ruler, I end up doing what
many people do and reach for the next best thing, which for me is a
sheet of A4 paper, 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

\[
\frac⚡💬 = \frac💬⚡
\]

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. They are precisely the
dimensions specified by the ISO standard for it. 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

\[
29.7 \, \mathrm{cm} \times 42.0 \, \mathrm{cm}.
\]

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 paper. 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?

So I was saying that we can bisect our paper like this and go all
the way to A\( \infty. \) That reminds me. Last night I was at a
bar in Hoxton and I saw an infinite number of mathematicians walk
in. The first one asked, “Sorry to bother you, but would it be
possible to have a sheet of A0 paper? I just need something to
scribble a few equations on.” The second one asked, “If you happen
to have one spare, could I please have an A1 sheet?” The third one
said, “An A2 would be perfectly fine for me, thank you.” Before the
fourth one could ask, the bartender disappeared into the back for a
moment and emerged with two sheets of A0 paper and said, “Right.
That should do it. Do know your limits and split these between
yourselves.”

In general, a sheet of A\( n \) paper has the dimensions

\[
2^{-(2n + 1)/4} \, \mathrm{m} \times
2^{-(2n – 1)/4} \, \mathrm{m}.
\]

If we plug in \( n = 4, \) we indeed get the dimensions of A4 paper:

\[
0.210 \, \mathrm{m} \times 0.297 \, \mathrm{m}.
\]

Measuring Stuff

Let us now return to the business of measuring things. As I
mentioned earlier, the dimensions of A4 are lodged firmly into my
memory. Getting hold of a sheet of A4 paper is rarely a challenge
where I live. I have accumulated a number of A4 paper stories over
the years. Let me share a recent one. I was hanging out with a few
folks of the nerd variety one afternoon when the conversation
drifted, as it sometimes does, to a nearby computer monitor that
happened to be turned off. At some point, someone confidently
declared that the screen in front of us was 27 inches. That sounded
plausible but we wanted to confirm it. So I reached for my trusted
measuring instrument: an A4 sheet of paper. What followed was
neither fast, nor especially precise, but it was more than adequate
for settling the matter at hand.

I lined up the longer edge of the A4 sheet with the width of the
monitor. One length. Then I repositioned it and measured a second
length. The screen was still sticking out slightly at the end. By
eye, drawing on an entirely unjustified confidence built from years
of measuring things that never needed measuring, I estimated the
remaining bit at about \( 1 \, \mathrm{cm}. \) That gives us a
width of

\[
29.7 \, \mathrm{cm} +
29.7 \, \mathrm{cm} +
1.0 \, \mathrm{cm}
=
60.4 \, \mathrm{cm}.
\]

Let us round that down to \( 60 \, \mathrm{cm}. \) For the height,
I switched to the shorter edge. One full \( 21 \, \mathrm{cm} \)
fit easily. For the remainder, I folded the paper parallel to the
shorter side, producing an A5-sized rectangle with dimensions \(
14.8 \, \mathrm{cm} \times 21.0 \, \mathrm{cm}. \) Using the \(
14.8 \, \mathrm{cm} \) edge, I discovered that it overshot the top
of the screen slightly. Again, by eye, I estimated the excess at
around \( 2 \, \mathrm{cm}. \) That gives us

\[
21.0 \, \mathrm{cm} +
14.8 \, \mathrm{cm}
-2.0 \, \mathrm{cm}
=
33.8 \, \mathrm{cm}.
\]

Let us round this up to \( 34 \, \mathrm{cm}. \) The ratio \( 60 /
34 \approx 1.76 \) is quite close to \( 16/9, \) a popular aspect
ratio of modern displays. At this point the measurements were
looking good. So far, the paper had not embarrassed itself.
Invoking the wisdom of the Pythagoreans, we can now estimate the
diagonal as

\[
\sqrt{(60 \, \mathrm{cm})^2 + (34 \, \mathrm{cm})^2}
\approx 68.9 \,\mathrm{cm}.
\]

Finally, there is the small matter of units. One inch is \( 2.54 \,
\mathrm{cm}, \) another figure that has embedded itself in my head.
Dividing \( 68.9 \) by \( 2.54 \) gives us roughly \( 27.2 \,
\mathrm{in}. \) So yes. It was indeed a \( 27 \)-inch display. My
elaborate exercise in showing off my A4 paper skills was now
complete. Nobody said anything. A few people looked away in
silence. I assumed they were reflecting. I am sure they were
impressed deep down. Or perhaps… no, no. They were definitely
impressed. I am sure.

Hold on. I think I hear another heckle. What is that? There are
mobile phone apps that can measure things now? Really? Right.
Security. Where’s security?