Nice! The author touches on the area properties and here's the most practical life hack derived from the standard I personally use. It uses the relationship between size and mass.

Because A0 is defined as having an area of exactly 1 square meter, the paper density (GSM or grams per square meter) maps directly to the weight of the sheet.

>A0 = 1 meter square.

>Standard office paper = 80 gsm

>Therefore, one sheet of A0 = 80 grams.

>Since A4 is 1/16th of an A0, a single sheet of standard A4 paper weighs 5 grams.

I rarely need to use a scale for postage. If I have a standard envelope (~5g) and 3 sheets of paper (15g), I know I'm at 20g total. It turns physical shipping logistics into simple integer arithmetic. The elegance of the metric system is that it makes the properties of materials discoverable through their definitions.

CGP Grey has a video [0] that goes into some, let's just say deeper, detail into metric paper that is well worth a watch if you haven't seen it.

0: https://www.youtube.com/watch?v=pUF5esTscZI

edit: fat-fingered CPG, thanks @ProllyInfamous

As a European living in North America I developed a weird cognitive dissonance. When I'm in North America the regular printer page (US Letter) seems too squat. When I'm in Europe the A4 page looks too svelte. I now need an in-between format for it to look "right".

I happened to be looking at ski boot fitting this morning and came across a web app from fischersports that allows you to measure your feet using your phones camera. Surprise surprise it uses a sheet of A4 paper.

App is about halfway down this page, https://www.fischersports.com/rc4-podium-rd-worldcup-strd/U0... under 'find your size' and is powered by https://volumental.com/

Fun article. I liked the bit about how the size of A0 can be uniquely determined from abstract constraints. But I'm not convinced that the "Measuring Stuff" section involves anything more than memorising the exact dimensions of A4. I don't see how it applies the stuff about preserved ratios.

Nitpick: typo in the dimensions for A3.

25. Of October 1786: Lichtenberg suggested his friend Beckmann a paper format in the aspect ration 1/√2.

»einen Bogen Papier zu finden, bey dem alle Formate … einander ähnlich wären. … Die kleine Seite des Rechtecks muß sich nämlich zu der großen verhalten wie 1:√2 oder wie die Seite des Quadrats zu seiner Diagonale. Die Form hat etwas angenehmes und vorzügliches vor der gewöhnlichen.«

Here’s a better tip to measure things without a proper measuring device: spread out your hand on a table and measure the distance between your pinky and thumb. Remember that. Now when you need to measure something just measure it in number of pinky-thumb-stretches. I can quickly get the dimensions to +/- an inch by doing a few quick walks with my hand.

Metric is beautiful.

I remember when I first got into metal work and wanted to get some tapping drills.

There are a plethora of standards when you start looking into it. For what I make though if I use metric I really only need one, ISO Coarse.

Metric is just well thought out and easier.

I use my fingers:

When I spread my index finger and middle finger, not entirely as far as they can go, but rather far, that's 10 centimetres.

Thumb to pinky is 22 centimetres. These two are often precise enough for me.

As an American I have done this with 8.5 x 11 "letter" paper. I wonder if there's some way one can take advantage of the special properties of A[n] paper.

Paper Towel stories:

I’ve started to determine the right package of paper towels to purchase according to the cents per square meter value. You can discern the quality of a deal at the grocery by referring to the ‘cents per X’ market located on price tag next to the marked price.

I’m beginning to turn sour on the ‘2 Jumbo-Mega-Rolls are the equivalent of 8 Super rolls’ scheme that’s en vogue. Are there retractable roll holders to accommodate for all of this?

It doesn’t help that many of these packages are priced and then marked down in ways to entice the buyer toward purchasing them instead of more reasonably priced and proportioned ones.

I've participated in (and now have been a ref for) a robotics competition, First Lego League, a couple of times. The robots can have a height of up to 30.5cm (12"), and that's usually quickly checked before each round.

Since there's pretty much always some A4 paper lying around near you, probably a copy of the rules, in most situations we just hold up a sheet (rolled up if you're feeling fancy) next to the robot. Much quicker than finding the ruler that's probably in use (or misplaced) somewhere else :D

Maths Youtuber Noel Friedrich recently made a video about A4 paper[0]. It turns out that since the ISO specification rounds the side lengths down to whole millimeters, with compounding errors, more than 2^10 A10s (smallest paper size in the standard) fit into one A0 (largest size in the standard).

[0] https://www.youtube.com/watch?v=zDKBCIMkDbw

Ha - I have made so many measurements using an A4 with great accuracy that this monitor-story might as well have been me :-)

I never understood the US paper size system while living there (or since...!), don't get me started with feet and inches and 16'ths etc - ISO, metric and base10 is just so much more logical and easy to use...

While this is nice, it is not inherently related to an A4 paper.

You could have memorized the length of a cheapo Bic pen if that is common in your area; Or a Parker or a Monte-Blanc if you carry one of those.

All recent iPhones (regular models since `03) have a width of 71.5mm. Remember that, and as long as someone near you has an iPhone, you are good to go using it as a ruler. (And people will definitely be, um, impressed).

We have in my kitchen several brands of small forks, all are 19.2mm (just checked. The large forks have a range of sizes). Next time I need to measure something I could just request a fork...

I use my phone when I want to measure stuff. Not an app, just the physical phone as a ruler. Almost always the dimensions of whatever phone I've got is published on the internet. It's a quick hack and better than carrying around A4 papers ;)

Pythagoras would appreciate her usage of his theorem, but I'd have just laid my papers diagonally across the screen to measure it directly without computing any square roots. Since I'm a yank, 11 + 11 + 5.5 would do quite nicely.

At one point on an international project I had to fly a box of UK A4 to the USA in my luggage so the Americans could check their software could cope with the different size. It did, but lugging it around was a pain - paper is heavy!

Yeah, I’ve used a sheet of paper as a ruler too...

As regards metric/A* paper sizes, it seems like just a coincidence that this scheme resulted in a standard size that is useful for everyday documents, since it only works for powers of 2 and starts with the definition of 1 square meter. If a meter were 1.5x smaller or larger, then I don’t think there would be a standard size that works so well.

EDIT: Being curious about this, I did some more reading, and discovered there is a “B” series of paper sizes that maintain the same ratio relationship, but are exactly in between all the A sizes! That’s useful.

Elegant, sure, but... fold a sheet of US Letter (8.5 x 11.0 inches) in half, and you're on your way to a pirate hat. Fold and pull it several more times, and you have a boat.

Some Moleskine cahier notebooks are wrapped in a paper sleeve that has a ruler printed on the back. Inches on one edge; centimetres on the other (a slight improvement for non-American users over Field Notes notebooks, with a ruler on the inside back cover that’s inches only - also the sleeve is rather longer, 28cm in fact).

I recently went down the A4 paper sizes rabbit hole and found out that there is an ISO standard for this 'ISO 216'

https://en.wikipedia.org/wiki/International_standard_paper_s...

Edit: while going through the article again, I noticed that's a brief mention of the ISO standard, which I missed in the first round.

I have heard German toilet paper is A6, but I never got to verify that. If true, it's one of the more German things ever.

21cm is close to 20 which means A5 has a side close to 10.

29.7 is close to 30. So why not use A4 sheets to install kitchen cabinets? A friend of mine advised me to buy a laser level. A tiny level was quite enough. A laser meter is nice though although I don't understand why it's inaccurate sometimes. Maybe it depends on the surface matter/paint.

A0 paper is for drafting. That's what people used for engineering drawings before we had AutoCAD with zoom.

Sqrt(2) being halfway between 1 and 2 multiplicatively, leads to interesting stuff. For example consider two integers A and B. They have a dual in the pair A+B and A-B. Well, not quite. You need to scale them down by 1/sqrt(2). If you do that to the duals, you get the originals again.

As the ratio used is a rational approximation of an irrational, I would guess that the ratio-preserving feature breaks down well before you get to atomic sizes, though I have neither proved that to be so nor figured out how to calculate the divergence.

If you have a Pro or Pro Max model of iPhone from the last several years, it has a LiDAR that allows the pre-installed Measure app to measure lengths/heights, etc., using the camera. Several higher end Android phones may also have the same.

Just a couple of days ago I used an A4 sheet to measure the area I had on countertop to see what size microwave I could fit in there.

Adds gsm issue to https://github.com/librasteve/raku-Physics-Unit/issues/32

Hint: The aspect ratio sqrt(2) should be quite convenient for foldable phones. Current ones seem to be more or less square if unfolded - what use case is that good for?

>Like most sensible people with a reasonable sense of priorities, I do not carry a ruler with me wherever I go.

Let me introduce to you: the IKEA paper tape measure, folded neatly in your wallet.

There's also the fact that if you don't have a convenient straight-edge around -- fold a sheet of paper, not too rough.

It's a good exercise in thinking, dhy that is so.

American: "Wow -- yet another example of practical math linked to everyday sources."

European: "Yes, but not your everyday sources."

1190mm (A0 height) / 841mm (A0 width) ⩰ √2

11" (US letter height) / 8.5" (US letter width) = 1.29411 ...

Countries using the Metric system: 192

Countries using the Imperial system: 3 (U.S., Liberia, Myanmar)

The U.S. rejection of the Metric system can be traced to Thomas Jefferson, who declared it "Too French."

There's also the fact that if you don't have a convenient straight-edge around -- fold a sheet of paper, not too rough.

What ... not everybody carries a small, 6' tape measure with them??? Surely I'm not THAT abnormal.

You can use objects of known length to measure other objects of unknown length. Am I a hacker, now?

Obligatory link to Markus Kuhn's excellent page on international/metric paper sizes: https://www.cl.cam.ac.uk/~mgk25/iso-paper.html

> We drink our tea from doughnuts.

Lol

Trying not to be too negative but this is quite a long blog post for what effectively just comes down to using a piece of paper to measure something, which I imagine lots of us have done in a pinch. Also, nit: one of the calculations on paper size seems duplicated due to a typo.