# Why is the A1 (paper) format 594×841 and not 594×840? [closed]

Why is the A1 (paper) format 594×841 mm and not 594×840 mm(so the double size of format A2)? Same goes with formats B, C, D, ....

• I'm voting to close this question as off-topic because this I nt about photography Nov 30, 2019 at 13:14

## 2 Answers

Let's take a look at the mathematics behind An paper sizing.

The aspect ratio is r = w/h where h is the longer side.

Now you want e.g. A1 to be half of A0, so the aspect ratio is r=(0.5*h)/w

We have the equation r = w/h = 0.5*h/w = 0.5/r or r^2 = 0.5

Then r is simply square root of 0.5 or in other words 0.70711.

Thus, the width (shorter side) is 0.70711 times height (longer side).

A0 paper size has area of 1 square meters. Thus, the longer side is h*h*r = 1 m2 or h*h = sqrt(2) m2 or h = sqrt(sqrt(2)) = 1.18920711500272 m. So, A0 paper size is 1.18920711500272 m x 0.840896415253715 m.

A1 paper size is then 0.840896415253715 m x 0.594603557501360 m.

A2 paper size is then 0.594603557501360 m x 0.420448207626857 m.

Rounding the numbers, we get 595 mm x 420 mm for A2.

Rounding the numbers, we get 841 mm x 595 mm for A1.

The reason it's 594 mm and not 595 mm is because the standard states:

Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the area of the preceding paper size and rounding down

Anyway, from Wikipedia's article on ISO 216 (the standard that defines the A and B paper sizes):

The tolerances specified in the standard are

• ±1.5 mm (0.06 in) for dimensions up to 150 mm (5.9 in),
• ±2 mm (0.08 in) for lengths in the range 150 to 600 mm (5.9 to 23.6 in) and
• ±3 mm (0.12 in) for any dimension above 600 mm (23.6 in).

So the difference is anyway within the specified tolerance.

• Gosh, thank you so so much! Very nicely explained, thank you! Nov 30, 2019 at 14:41
• BTW, the reason A2 is rounded from 594.603... to 594 (and not 595) is also in the WP article you cite: "Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the area of the preceding paper size and rounding down"
– scottbb
Nov 30, 2019 at 14:55
• Wikipedia notwithstanding, the standard ISO 216:2007 (the current version; but what follows is also true for the 1975 version) does not spell out what procedure was used to convert the irrational numbers (produced by the underlying principles) into integer numbers of millimeters it finally lists. If there are separate standards for rounding numbers, ISO 216 doesn't mention them. The best one can do is back-engineer what rounding procedure ISO had in mind, as was done e.g. here. Apr 16, 2023 at 0:18

Because the standard is based on the square root of two — an irrational number — but each size is rounded to the nearest whole millimeter.

The standard also allows for slight variation from the nominal number.

In both theory and most practical use, A1 is exactly double A2. If you need precision beyond that, you should cut your own paper.

• So then, for example, if I know the size of A2, how can I get the size of A1? Nov 30, 2019 at 13:24
• @Vaninyaa Generally, the long edge of A2 is the same dimension as the short edge of A1. A1's long edge is roughly sqrt(2) times A2's long edge. But technically, everything is defined from A0, and A1...A10 are derived from the definition of A0.
– scottbb
Nov 30, 2019 at 15:02