That a DIN A4 sheet measures 210 x 297 mm has little chance and a lot of mathematics

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We have been using sheets in DIN A4 format for decades, and surely many of you have asked yourselves the same question of when and when: Why does a DIN A4 sheet measure 210 x 297 mm? Why not other more “round” dimensions?

In fact, one might think that it would be much better to end up using sheets of, for example, 200 x 300 mm to make everything easier. Perhaps it would be to remember those dimensions, but then the DIN A4 sheet would not be so perfect. And it is thanks to mathematics.

The magic of maintaining “aspect ratio”

If one takes any piece of paper that is not a conventional folio in some DIN format, one will find a curious situation. You will be able to fold it without problems, but doing so those halves will no longer have the format of the original paper. They will be more rectangular or more square, but will not retain the “aspect ratio” of the original paper.

That is precisely the secret of the DIN A4 format, and that is where the math comes in. in 1786 a letter from the German academic Georg Christoph Lichtenberg to Johann Beckmann —who coined the word ‘technology’, that’s nothing— formulated the idea of ​​using a paper format that could be preserved by folding it (or expand it proportionally).

It was not until the early 20th century that Germany managed to standardize the idea. Now that standard is known as ISO216and defines the international standard paper size for most of the world. How was that standard defined?


Well, with only one objective: that the aspect ratio be maintained, and this is where a simple mathematical operation allowed to solve the problem. As the mathematician explained Ben Sparks, one can draw a rectangle with aspect ratio x:1. If one divides the rectangle in half, the new rectangle will have an aspect ratio of 1:x/2.

If you apply math and want both aspect ratios to be the same, just solve the equation x/1 = 1/(x/2), which at the end causes the result to be obtained that x = √2.

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So that is the only solution to keep the aspect ratio. Since there is not a pair of integers that allow to obtain an aspect ratio √2, approximations are used. Some approximations that, yes, start from an almost perfect number.

Thus, the paper A0 (DIN A0) uses that aspect ratio and is 1 m² in area. Or almost, because its dimensions (1,189 x 841 mm) are quite close to that “round” area (999,949 mm²).


Source: Wikipedia.

From there we fold the A0 several times (one, two, three, four, five, …) to successively and respectively obtain paper A1, A2, A3, A4 or A5, etc., which have dimensions that are half of the previous format, and that, of course, maintain aspect ratio. Magic. Or math, rather.

There is some other curiosity associated with that aspect ratio. The first, that paper weight can be easily calculated: If 80 gsm (grams per square meter) paper is used, an A0 sheet will weigh exactly 80 grams. An A4 sheet with that density will weigh 5 g (because we have folded —divided— the A0 four times).

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These rotrings advance in thickness by multiplying the previous one by 1.4. They are not exact, but the relationship remains fairly stable.

The second is that the thicknesses of technical markers are also usually increased, maintaining this ratio of √2, or what is (almost) the same, 1.4. that way the following thickness of a marker pen will be suitable to draw on the next size of paper.

We have been using sheets in DIN A4 format for decades, and surely many of you have asked yourselves the…

We have been using sheets in DIN A4 format for decades, and surely many of you have asked yourselves the…

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