Paper Sizes

Did you know that international paper sizes (like A3, A4, etc) are based on the square root of 2?

Because the square root of 2 has this cool property:

1 × √2 × √2 = 2

Which lets us have this:

paper sizes root 2

So we can have sheets that have exactly the same proportions (their ratio of side lengths are the same) and also fit in each other perfectly:

paper sizes A4 A3
Two A4s make an A3
and have the same proportions

This makes things really efficient:

And because they have the same proportions, any artwork or document can be resized to fit on any sheet:

paper size resize

Another benefit is that you can print something out at 70% size and fit 2 pages side-by-side on just one sheet like this:

paper size resize

Why 70%? Because 1/√2 = 0.7071... which is close to 70%

A similar enlargement is √2 = 1.4142... which is close to 140%


The popular A4 size is 210 mm wide by 297 mm high:

paper sizes

With a width of 210 the height is: 210 × √2 ≈ 297


Here are all the sizes cut from an A0 sheet (which has an area of 1.0 m2):

paper sizes

Lastly here are the official sizes:

size mm × mmabout the size of a area
A0841 × 1189table top 1.0 m2
A1594 × 841  0.5 m2
A2420 × 594monitor 0.25 m2
A3297 × 420  0.125 m2
A4210 × 297writing sheet 0.0624 m2
A5148 × 210  0.0311 m2
A6105 × 148  0.0155 m2
A774 × 105note 0.00777 m2
A852 × 74  0.003848 m2
A937 × 52  0.001924 m2
A1026 × 37stamp 0.000962 m2

Note: you can think of the "A-number" as how many folds (or cuts-in-half) away from an A0 we are. So an A3 needs 3 folds of an A0, and so is ½×½×½ = 1/8th the size.