More About Paper Sizes

The ISO paper size concept

In the ISO paper size system, the height-to-width ratio of all pages is the square root of two (1.4142 : 1). In other words, the width and the height of a page relate to each other like the side and the diagonal of a square. This aspect ratio is especially convenient for a paper size. If you put two such pages next to each other, or equivalently cut one parallel to its shorter side into two equal pieces, then the resulting page will have again the same width/height ratio.


The ISO paper sizes are based on the metric system. The square-root-of-two ratio does not permit both the height and width of the pages to be nicely rounded metric lengths. Therefore, the area of the pages has been defined to have round metric values. As paper is usually specified in g/m², this simplifies calculation of the mass of a document if the format and number of pages are known.
ISO 216 defines the A series of paper sizes based on these simple principles:

• The height divided by the width of all formats is the square root of two (1.4142).
• Format A0 has an area of one square meter.
• Format A1 is A0 cut into two equal pieces. In other words, the height of A1 is the width of A0 and the width of A1 is half the height of A0.
• All smaller A series formats are defined in the same way. If you cut format An parallel to its shorter side into two equal pieces of paper, these will have format A(n+1).
• The standardized height and width of the paper formats is a rounded number of millimeters.

For applications where the ISO A series does not provide an adequate format, the B series has been introduced to cover a wider range of paper sizes. The C series of formats has been defined for envelopes.

• The width and height of a Bn format are the geometric mean between those of the An and the next larger A(n−1) format. For instance, B1 is the geometric mean between A1 and A0, that means the same magnification factor that scales A1 to B1 also scales B1 to A0.
• Similarly, the formats of the C series are the geometric mean between the A and B series formats with the same number. For example, an (unfolded) A4 size letter fits nicely into a C4 envelope, which in turn fits as nicely into a B4 envelope. If you fold this letter once to A5 format, then it will fit nicely into a C5 envelope.
• B and C formats naturally are also square-root-of-two formats.

Note: The geometric mean of two numbers x and y is the square root of their product, (xy)^1/2, whereas their arithmetic mean is half their sum, (x+y)/2. For example, the geometric mean of the numbers 2 and 8 is 4 (because 4/2 = 8/4), whereas their arithmetic mean is 5 (because 5−2 = 8−5). The arithmetic mean is half-way between two numbers by addition, whereas the geometric mean is half-way between two numbers by multiplication.

By the way: The Japanese JIS P 0138-61 standard defines the same A series as ISO 216, but a slightly different B series of paper sizes, sometimes called the JIS B or JB series. JIS B0 has an area of 1.5 m², such that the area of JIS B pages is the arithmetic mean of the area of the A series pages with the same and the next higher number, and not as in the ISO B series the geometric mean. For example, JB3 is 364 × 515, JB4 is 257 × 364, and JB5 is 182 × 257 mm. Using the JIS B series should be avoided. It introduces additional magnification factors and is not an international standard. 

The ISO standard paper size system covers a wide range of formats, but not all of them are widely used in practice. Among all formats, A4 is clearly the most important one for daily office use. Some main applications of the most popular formats can be summarized as:

A0, A1	technical drawings, posters
A1, A2	flip charts
A2, A3	drawings, diagrams, large tables
A4	letters, magazines, forms, catalogs, laser printer and copying machine output
A5	note pads
A6	postcards
B5, A5, B6, A6	books
C4, C5, C6	envelopes for A4 letters: unfolded (C4), folded once (C5), folded twice (C6)
B4, A3	newspapers, supported by most copying machines in addition to A4
B8, A8	playing cards

You are in a library and want to copy an article out of a journal that has A4 format. In order to save paper, you want copy two journal pages onto each sheet of A4 paper. If you open the journal, the two A4 pages that you will now see together have A3 format. By setting the magnification factor on the copying machine to 71% (that is sqrt(0.5)), or by pressing the A3→A4 button that is available on most copying machines, both A4 pages of the journal article together will fill exactly the A4 page produced by the copying machine. One reproduced A4 page will now have A5 format. No wasted paper margins appear, no text has been cut off, and no experiments for finding the appropriate magnification factor are necessary. The same principle works for books in B5 or A5 format.


Copying machines designed for ISO paper sizes usually provide special keys for the following frequently needed magnification factors:

71%	sqrt(0.5)	A3 → A4
84%	sqrt(sqrt(0.5))	B4 → A4
119%	sqrt(sqrt(2))	A4 → B4 (also B5 → A4)
141%	sqrt(2)	A4 → A3 (also A5 → A4)

The magnification factors between all A sizes:

To see more:http://www.cl.cam.ac.uk/~mgk25/iso-paper.html