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Comment by jstanley | Hacker News Reader

The article makes these 2 contradictory claims:

> I can tell you now it is precisely 210mm by 297mm

> So the only ratio that has this important property is the square root of 2, famously – and ironically in this case – not a ratio

I suspect either:

- A4 paper is not precisely 210x297 mm, or

- the precision of the ratio has been sacrificed in order to allow integer numbers; but if we're willing to make that sacrifice, why not round up to 212x300 mm? and then we can have the nice round number that the classmate wanted in the first place, with a bonus that both dimensions are even-numbered so can be halved with mm-graduated rulers.

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I was wondering that too! I got to the end and still no explanation of 210×297mm.

Well, it turns out that it's based on the fact that A0 is defined as having an area of ~1m², then each subsequent size halves the area. So A4 has dimensions that give an aspect ratio √2 and an area of ~1/16m².

    A0: 1,189 mm × 841 mm = 999,949mm² ≈ 1m²
    ...
    A4: 297 mm × 210 mm = 62,370mm² ≈ 1/16m²

See https://en.wikipedia.org/wiki/Paper_size#A_series :

"The A0 base size is defined as having an area of 1 m² ... rounded to the nearest millimetre, 1,189 mm × 841 mm".

That's a beautifully short definition, I think.

"Base-size has 1sqm area. Halving does not change ratio of the sides."

You may have skimmed over this bit in the article then:

"the international A-size system now starts with A0 paper, with sides in the correct 2‾√ ratio, but with exactly 1m2 of area, or as near as you can get with whole millimetre side lengths (1189mm 841mm, to be precise). Then folding or cutting it a few times until it is conveniently desk/folder sized gets you to A4, hence the ‘4’ bit in the name."

Yes I did! Thanks for pointing it out and very politely too. :)

"Unnecessary accuracy is technical incompetence" - Engineering saying. You need accuracy to serve some purpose. Too much is inefficient and expensive. Too little does not work. 212x300 would not serve the purpose. Try folding and you see how bad it is.

A4 paper is precisely 210x297mm (according to the standard), so the aspect ratio deviates from 1:sqrt(2) a little. Following that aspect ratio, the size would be 210x296.985..mm, so the rounding is really very slight.

How exactly is it defined in the standard? Is there a formula? Or just a list of sizes for A0, A1, etc.?

I wanted to read the standard myself, but CHF 58 is a bit too much:

https://www.iso.org/standard/36631.html

   y : x = √2 : 1 = 1,414

http://www.ikserang.com/qa/ISO_PAPER_BOARD/standards/en/iso_...

This seems to be a sample containing the relevant parts. https://cdn.standards.iteh.ai/samples/36631/c0883203ea25445c...

> why not round up to 212x300 mm?

Because the A standard's baseline is A0 which has exactly 1m^2 area. So that nice rounding did happen, just at the baseline scale, and on area. The B series is a tad larger, B0 happens to exactly be 1,000 × 1,414 mm.

Which also makes it quite nice to know how much a sheet of paper weighs. At 80 g/m² you'll get 5 g (80 / 2⁴) for A4. Handy for quickly determining how much postage to put on an envelope based on how many pages you put inside it.

I love describing 41,4% as tad. Then again I suppose concept of comparing areas is bit harder.

Typically, paper comes in A0 size (841x1188mm) which is then cut in half X times to get the desired paper size, which is either eyeballed against a ruler, or measured by laser.

If the operator is skilled, or the machine is well-calibrated most A4 paper is therefore exactly 297mm tall, and 210.25mm wide (so slightly wider than the standard)

1) The integer number is A0 being 1 square meter of area.

2) Rounding to whole centimeters, or an even number of millimeters, is too coarse.

I guess the second claim is referring to the fact that the square root of 2 is an irrational number, so it can’t be expressed as a ratio of integers anyway.

I'd guess because 210x297 is closer to 1/16 of a m^2 than 212x300 is.

A0 has an area of 1 m² and the long edge is sqrt(2) of the short one. This fixes the edge lengths to 841 mm x 1189 mm (rounded to the nearest mm). A4 is obtained by halving A0 four times.

If you change the size of A4 you have to change the size of all the other A- sizes too.

A0 is anchored at 1 metre squared, so everything else follows from that.

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