The Rationale Behind Standard Paper Sizes Like A4 and A3
The A series of paper sizes are designed so that when you cut one in half, you get two pieces of the next-smallest-size, and every size has height and width in the same proportion. A little math reveals that one can achieve this by having the height and width in the ratio sqrt(2):1, or approximately 1.414:1.
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In the case of the A series of paper sizes, an A0 piece of paper is exactly 1 square metre, requiring width x width x 1.414 = 1, which gives a width of 84.1cm and height of 118.9cm (to the nearest mm).
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Another interesting consequence is that it becomes easy to calculate the weight of single sheets. Standard photocopy paper is usually 80gsm (grams per square metre), thus an A0 sheet, being 1 square metre, weighs 80g. An A1 weighs 40g, A2 is 20g, A3 is 10g, and A4 is 5g. (And so on.)
This “halving” situation is also fantastic for envelope sizes. A C4 envelope will hold a sheet of A4 paper. A C5 will hold A5… or A4 folded in half. A C6 will hold A6, or an A4 folded in quarters.
Because the ratio of the sides of all A size papers is root 2, or 1.41, it’s always the case that expanding a document from one size to the next largest needs a factor of 141% dialled into your photocopier or typed into Illustrator or whatever. Reducing from one A size to the next smallest requires a factor of 1 over root 2, or 0.71, or 71%.
Say, for example, you have a map printed on A1 paper and the map is at a scale of 1:1,000. If you want to print the map on smaller paper, because of the ratio of the paper sizes, you can drop the paper two sizes (to A3) and the scale will reduce to 1:2,000.
See also Wikipedia’s articles about paper size and ISO 216.
