How A4 Paper Got Its Size

We had to make a chart in school, which had to be printed on A3 paper, which we had run out of. That’s when someone said, “Let’s print them on two A4s and stick them together”.

 

This got me thinking more deeply about how A4 was half of A3, and double of A5 – clearly this scaling symmetry was by design. Some quick research revealed that this scaling was used in Germany as early as in the early 1900s, when engineers wanted paper sizes that would scale neatly, whereby when they were folded or cut in half, the shape stayed the same.

 

Most rectangles don’t necessarily behave this way. If you take a random sheet of paper and fold it in half along the longer side, the new rectangle will look wider, i.e. the aspect ratio (the ratio of the length of the rectangle to its breadth) changes. But someone had the genius idea to think about a ratio that doesn’t change when the sheet is folded or cut, and here’s the maths behind it.

 

Suppose the paper has l and b as the length and breadth (width) respectively. The aspect ratio is l : b. If we cut this sheet in half along its longer side, the new sheet will have sides of length l / 2 and b. Now, for the new rectangle to have the same aspect ratio as the original, i.e. for the ratio of length to breadth to remain unchanged, when b becomes the length, and l / 2 becomes the breadth, mathematically l / b must be equal to b ÷ (l / 2)

If   l  =     b     , as stated above

     b      (l /2)   

∴ l² / 2 = b²

∴ l² / b² = 2

∴ l / b = √2

 

In other words, the ratio of the length to the breadth will be 1: √2.

 

Intuitively, the way this works is that since √2 / 2 = 1 / √2, when we halve the length of the original rectangle and now take that as the breadth, our new rectangle still has the same length to breadth ratio since the original aspect ratio was √2 : 1 and the new aspect ratio is 

1 : 1 / √2.

 

When this system was first created, A0 was taken as the starting point, whose area was exactly 1m², and whose sides were in this 1:√2 ratio. From there, each successive paper size in the A series halved in area but had the same ratio of its dimensions. These are all sizes defined by ISO 216. The ISO standard also has B and C series of paper dimensions, all of which are based on the length to breadth ratio of √2.

A series paper sizes
Size	Breadth (mm)	Length (mm)	Area (m²)	Relation
A0	841	1189	1	Base
A1	594	841	1/2	Half of A0
A2	420	594	1/4	Half of A1
A3	297	420	1/8	Half of A2
A4	210	297	1/16	Half of A3
A5	149	210	1/32	Half of A4

This is how we get the seemingly unusual dimensions of 210 x 297 mm for A4 paper!