International standard paper sizes


ISO 216 is an international standard for paper sizes, used around the world except in North America and parts of Latin America. The standard defines the "A", "B" and "C" series of paper sizes, which includes the A4, the most commonly available paper size worldwide. Two supplementary standards, ISO 217 and ISO 269, define related paper sizes.
All ISO 216, ISO 217 and ISO 269 paper sizes have the same aspect ratio, square root of 2|, within rounding to millimetres. This ratio has the unique property that when cut or folded in half widthways, the halves also have the same aspect ratio. Each ISO paper size is one half of the area of the next larger size in the same series.

Dimensions of A, B and C series

History

The oldest known mention of the advantages of basing a paper size on an aspect ratio of is found in a letter written on 25 October 1786 by the German scientist Georg Christoph Lichtenberg to Johann Beckmann, both at the University of Göttingen. Early variants of the formats that would become ISO paper sizes A2, A3, B3, B4, and B5 then evolved in France, where they were listed in a 1798 French law on taxation of publications that was based in part on page sizes.
Image:Comparison_paper_sizes.svg|thumb|Comparison of A4 and C4 sizes with some similar paper and photographic paper sizes
Searching for a standard system of paper formats on a scientific basis at the Bridge association, as a replacement for the vast variety of other paper formats that had been used before, in order to make paper stocking and document reproduction cheaper and more efficient, in 1911 Wilhelm Ostwald proposed, over a hundred years after the 1798 French law, a global standarda world format for paper sizes based on the ratio, referring to the argument advanced by Lichtenberg's 1786 letter, but linking this to the metric system using as the width of the base format. argued in a long article published in 1918, that a firm basis for the system of paper formats, which deal with surfaces, ought not be the length but the area; that is, linking the system of paper formats to the metric system using the square metre rather than the centimetre, constrained by and area square metre, where is the length of the shorter side and is the length of the longer side, for the second equation both in metres. Porstmann also argued that formats for containers of paper, such as envelopes, should be 10% larger than the paper format itself.
In 1921, after a long discussion and another intervention by Porstmann, the Standardisation Committee of German Industry, which is the German Institute for Standardisation today, published German standard DI Norm 476 the specification of four series of paper formats with ratio, with series A as the always preferred formats and basis for the other series. All measures are rounded to the nearest millimetre. A0 has a surface area of up to a rounding error, with a width of and height of, so an actual area of ; A4 is recommended as standard paper size for business, administrative and government correspondence; and A6 for postcards. Series B is based on B0 with width of, C0 is, and D0. Series C is the basis for envelope formats.
The DIN paper-format concept was soon introduced as a national standard in many other countries, for example, Belgium, Netherlands, Norway, Switzerland, Sweden, Soviet Union, Hungary, Italy, Finland, Uruguay, Argentina, Brazil, Spain, Austria, Romania, Japan, Denmark, Czechoslovakia, Israel, Portugal, Yugoslavia, India, Poland, United Kingdom, Venezuela, New Zealand, Iceland, Mexico, South Africa, France, Peru, Turkey, Chile, Greece, Zimbabwe, Singapore, Bangladesh, Thailand, Barbados, Australia, Ecuador, Colombia and Kuwait.
It finally became both an international standard as well as the official United Nations document format in 1975, and it is today used in almost all countries in the world, with the exception of several countries in the Americas.
In 1977, a large German car manufacturer performed a study of the paper formats found in their incoming mail and concluded that out of 148 examined countries, 88 already used the A series formats.

Advantages

The main advantage of this system is its scaling. Rectangular paper with an aspect ratio of has the unique property that, when cut in two across the midpoints of the longer sides, each half has the same aspect ratio as the whole sheet before it was divided. Equivalently, if one lays two same-sized sheets of paper with an aspect ratio of side by side along their longer side, they form a larger rectangle with the aspect ratio of and double the area of each individual sheet.
The ISO system of paper sizes exploits these properties of the aspect ratio. In each series of sizes, the largest size is numbered 0, and each successive size has half the area of the preceding sheet and can be cut by halving the length of the preceding size sheet. The new measurement is rounded down to the nearest millimetre. A folded brochure can be made by using a sheet of the next larger size. An office photocopier or printer can be designed to reduce a page from A4 to A5 or to enlarge a page from A4 to A3. Similarly, two sheets of A4 can be scaled down to fit one A4 sheet without excess empty paper.
This system also simplifies calculating the weight of paper. Under ISO 536, paper's grammage is defined as a sheet's mass in grams per area in square metres. One can derive the weight of other sizes by arithmetic division. A standard A4 sheet made from paper weighs, as it is of an A0 page. Thus the weight, and the associated postage rate, can be approximated easily by counting the number of sheets used.
ISO 216 and its related standards were first published between 1975 and 1995:
  • ISO 216:2007, defining the A and B series of paper sizes
  • ISO 269:1985, defining the C series for envelopes
  • ISO 217:2013, defining the RA and SRA series of raw paper sizes

    Properties

A series

Paper in the A series format has an aspect ratio of . A0 is defined so that it has an area of before rounding to the nearest. Successive paper sizes in the series are defined by halving the area of the preceding paper size and rounding down, so that the long side of is the same length as the short side of An. Hence, each next size is nearly exactly half the area of the prior size. So two A2 pages fit together over an A1 page, an A3 page is half an A2 page, A4 is half an A3 and so on.
The most used of this series is the A4 paper size, which is and thus almost exactly in area. For comparison, the letter paper size commonly used in North America is about wider and shorter than A4. Then, the size of A5 paper is half of A4, i.e. × .
The geometric rationale for using the square root of 2 is to maintain the aspect ratio of each subsequent rectangle after cutting or folding an A-series sheet in half, perpendicular to the larger side. Given a rectangle with a longer side, x, and a shorter side, y, ensuring that its aspect ratio,, will be the same as that of a rectangle half its size,, which means that, which reduces to ; in other words, an aspect ratio of.
Any paper can be defined as, where
Therefore
etc.

B series

The B series is defined in the standard as follows: "A subsidiary series of sizes is obtained by placing the geometrical means between adjacent sizes of the A series in sequence." The use of the geometric mean makes each step in size: B0, A0, B1, A1, B2... smaller than the previous one by the same factor. As with the A series, the lengths of the B series have the ratio, and folding one in half gives the next in the series. The shorter side of B0 is exactly 1 metre.
There is also an incompatible Japanese B series which the JIS defines to have 1.5 times the area of the corresponding JIS A series. Thus, the lengths of JIS B series paper are ≈ 1.22 times those of A-series paper. By comparison, the lengths of ISO B series paper are ≈ 1.19 times those of A-series paper.
Any paper can be defined as, where
Therefore
etc.

C series

The C series formats are geometric means between the B series and A series formats with the same number. The width to height ratio of C series formats is as in the A and B series. A, B, and C series of paper fit together as part of a geometric progression, with ratio of successive side lengths of, though there is no size half-way between Bn and : A4, C4, B4, "D4", A3,...; there is such a D-series in the Swedish extensions to the system. The lengths of ISO C series paper are therefore ≈ 1.09 times those of A-series paper.
The C series formats are used mainly for envelopes. An unfolded A4 page will fit into a C4 envelope. Due to same width to height ratio, if an A4 page is folded in half so that it is A5 in size, it will fit into a C5 envelope.
Any paper can be defined as, where
Therefore
etc.

Tolerances

The tolerances specified in the standard are:
  • ±1.5 mm for dimensions up to 150 mm,
  • ±2.0 mm for dimensions in the range 150 to 600 mm, and
  • ±3.0 mm for dimensions above 600 mm.
These are related to comparison between series A, B and C.

Application

The ISO 216 formats are organized around the ratio 1:; two sheets next to each other together have the same ratio, sideways.
In scaled photocopying, for example, two A4 sheets reduced to A5 size fit exactly onto one A4 sheet, and an A4 sheet in magnified size onto an A3 sheet; in each case, there is neither waste nor want.
The principal countries not generally using the ISO paper sizes are the United States and Canada, which use North American paper sizes. Although many Latin American countries have also officially adopted the ISO 216 paper format, Mexico, Panama, Peru, Colombia, the Philippines, and Chile also use mostly U.S. paper sizes.
Rectangular sheets of paper with the ratio 1: are popular in paper folding, such as origami, where they are sometimes called "A4 rectangles" or "silver rectangles". In other contexts, the term "silver rectangle" can also refer to a rectangle in the proportion 1:, known as the silver ratio.