What is the rationale behind standard paper sizes like A4 and A3?
10 Answers
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.
So once you've decided that this is the ratio you want for your page, how do you decide the absolute size?
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).
An A1 piece of paper has a length that's the same as the width of A0, or 84.1cm, and a width of 84.1cm / sqrt (2) = 59.5cm. Which you'll notice is half the height of the A0 size (118.9cm).
These relationships hold true going all the way down to A10:
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It happens that of the A series, A4 is the one that's closest to the traditional size used for correspondence, and this has therefore become the de facto standard paper size in the metric world (i.e. pretty much globally except the United States :) ).
One fairly obvious consequence of this is that one can make a booklet of a particular size, from a sheet of the next-largest-size folded in half. So A3 sheets of paper can be folded to create A4-size booklets.
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.)
So once you've decided that this is the ratio you want for your page, how do you decide the absolute size?
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).
An A1 piece of paper has a length that's the same as the width of A0, or 84.1cm, and a width of 84.1cm / sqrt (2) = 59.5cm. Which you'll notice is half the height of the A0 size (118.9cm).
These relationships hold true going all the way down to A10:
It happens that of the A series, A4 is the one that's closest to the traditional size used for correspondence, and this has therefore become the de facto standard paper size in the metric world (i.e. pretty much globally except the United States :) ).
One fairly obvious consequence of this is that one can make a booklet of a particular size, from a sheet of the next-largest-size folded in half. So A3 sheets of paper can be folded to create A4-size booklets.
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.)
In addition to the A series sizes, there are two other ISO series- B and C which too are based on the same proportion (hence foldable exactly to the next size in each series)
B series: This was added to provide wider range of paper sizes. It is based on geometric mean between two consecutive A series sheets. i.e., B4 is between A3 and A4 size.
Consequently, if we take A and B series sequentially as per their sizes, i.e, B0, A0, B1, A1, B2, each size is smaller than the next by an equal scale.
Interestingly, while in A series the area of an A0 sheet is 1 sqm, in B series, the shorter side of B0 is exactly 1 m.
B series is used for certain special purposes, such as Books (B5), passports (B7) and envelopes for lesser used C series paper sizes.
C series: This is mainly used for envelopes for the most commonly used A series paper sizes, and it is based on geometric mean between two consecutive A and B series sheets of the same number. i.e, C4 is between B4 and A4 size
So in terms of sequence of sizes you have B0, C0, A0, B1, C1, A1...etc
All the above, i.e, A, B & C series rectangles follow the Silver ratio (la golden ratio) and accordingly they are also known as silver rectangles
B series: This was added to provide wider range of paper sizes. It is based on geometric mean between two consecutive A series sheets. i.e., B4 is between A3 and A4 size.
Consequently, if we take A and B series sequentially as per their sizes, i.e, B0, A0, B1, A1, B2, each size is smaller than the next by an equal scale.
Interestingly, while in A series the area of an A0 sheet is 1 sqm, in B series, the shorter side of B0 is exactly 1 m.
B series is used for certain special purposes, such as Books (B5), passports (B7) and envelopes for lesser used C series paper sizes.
C series: This is mainly used for envelopes for the most commonly used A series paper sizes, and it is based on geometric mean between two consecutive A and B series sheets of the same number. i.e, C4 is between B4 and A4 size
So in terms of sequence of sizes you have B0, C0, A0, B1, C1, A1...etc
All the above, i.e, A, B & C series rectangles follow the Silver ratio (la golden ratio) and accordingly they are also known as silver rectangles
I think its so fascinating and clever the way it was done.
for the academic history, i think wikipedia page is good ISO 216
I wrote a blog post on my site about finding the GSM of a sheet of paper if anyone is interested, but its basically same as whats mentioned above.
How to find out the GSM of a piece of paper
for the academic history, i think wikipedia page is good ISO 216
I wrote a blog post on my site about finding the GSM of a sheet of paper if anyone is interested, but its basically same as whats mentioned above.
How to find out the GSM of a piece of paper
An interesting byproduct of using the A series of paper is that when you work with maps that are printed on such paper, the scales of the maps are easy to convert when changing paper sizes.
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.
Likewise, if you have a map printed on A4 paper and the scale of the map is 1:50,000 and you want to print the map at a scale of 1:25,000, you can print it on A2 paper.
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.
Likewise, if you have a map printed on A4 paper and the scale of the map is 1:50,000 and you want to print the map at a scale of 1:25,000, you can print it on A2 paper.
Aakash - The short answer is no. Long answer: The golden ratio is defined as such: two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Or in symbols, (a+b)/a = a/b. The two numbers are different; as already stated the A4 proportions are 1.41 to 1, whereas the golden ratio is roughly 1.62 to 1. The golden ratio is called that because the ancient Greeks thought (and many still think) that it is the ideal ratio, aesthetically, for such things as the shapes of doorways.
