Wikipedia

# ISO 216

Visualization with paper sizes in formats A0 to A8, exhibited at the science museum CosmoCaixa Barcelona
An A4 paper sheet folded into two A5 size pages

ISO 216 is an international standard for paper sizes, used across the world except in North America and parts of Latin America. The standard defines the "A", "B" and "C" series of paper sizes, including A4, the most commonly available paper size worldwide. Two supplementary standards, ISO 217 and ISO 269, define related paper sizes; the ISO 269 "C" series is commonly listed alongside the A and B sizes.

All ISO 216, ISO 217 and ISO 269 paper sizes (except some envelopes) have the same aspect ratio, 2:1, 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.[1]

## Dimensions of A, B and C series

ISO paper sizes in millimetres and in inches
Size A series formats B series formats C series formats
mm inches mm inches mm inches
0 0841 × 1189 33.1 × 46.8 1000 × 1414 39.4 × 55.7 0917 × 1297 36.1 × 51.1
1 0594 × 0841 23.4 × 33.1 0707 × 1000 27.8 × 39.4 0648 × 0917 25.5 × 36.1
2 0420 × 0594 16.5 × 23.4 0500 × 0707 19.7 × 27.8 0458 × 0648 18.0 × 25.5
3 0297 × 0420 11.7 × 16.5 0353 × 0500 13.9 × 19.7 0324 × 0458 12.8 × 18.0
4 0210 × 0297 08.3 × 11.7 0250 × 0353 09.8 × 13.9 0229 × 0324 09.0 × 12.8
5 0148 × 0210 05.8 × 08.3 0176 × 0250 06.9 × 09.8 0162 × 0229 06.4 × 09.0
6 0105 × 0148 04.1 × 05.8 0125 × 0176 04.9 × 06.9 0114 × 0162 04.5 × 06.4
7 0074 × 0105 02.9 × 04.1 0088 × 0125 03.5 × 04.9 0081 × 0114 03.2 × 04.5
8 0052 × 0074 02.0 × 02.9 0062 × 0088 02.4 × 03.5 0057 × 0081 02.2 × 03.2
9 0037 × 0052 01.5 × 02.0 0044 × 0062 01.7 × 02.4 0040 × 0057 01.6 × 02.2
10 0026 × 0037 01.0 × 01.5 0031 × 0044 01.2 × 01.7 0028 × 0040 01.1 × 01.6
Comparison of ISO 216 paper sizes between A4 and A3 and Swedish extension SIS 014711 sizes

## History

The oldest known mention of the advantages of basing a paper size on an aspect ratio of 2 is found in a letter written on 25 October 1786 by the German scientist Georg Christoph Lichtenberg to Johann Beckmann.[2]

The formats that became ISO paper sizes A2, A3, B3, B4, and B5 were developed in France. They were listed in a 1798 law on taxation of publications that was based in part on page sizes.[3]

Comparison of A4 (shaded grey) and C4 sizes with some similar paper and photographic paper sizes

Searching for a standard system of paper formats on a scientific basis by the association Die Brücke – Internationales Institut zur Organisierung der geistigen Arbeit (The Bridge – International institute to organise intellectual work), 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, Wilhelm Ostwald proposed in 1911, over a hundred years after the “Loi sur le timbre”, a Weltformat (world format) for paper sizes based on the ratio 1:2, referring to the argument advanced by Lichtenberg 1786 letter, and linking this to the metrical system by using 1 centimetre as the width of the base format. W. Porstmann argued in a long article published in 1918, that a firm basis for the system of paper formats, which deal with surfaces, could not be the length, but the surface, i.e. linking the system of paper formats to the metrical system of measures by the square metre, using the two formulae of x / y = 1:2 and x × y = 1. Porstmann also argued that formats for containers of paper like envelopes should be 10% larger than the paper format itself.

In 1921, after a long discussion and another intervention by W. Porstmann, the Normenausschuß der deutschen Industrie (NADI, "Standardisation Committee of German Industry", today Deutsches Institut für Normung or short DIN) published German standard DI Norm 476 the specification of 4 series of paper formats with ratio 1:2, 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 1 square metre up to a rounding error, with a width of 841 mm and height of 1189 mm, so an actual area of 0.999949 m2, and A4 recommended as standard paper size for business, administrative and government correspondence and A6 for postcards. Series B is based on B0 with width of 1 metre, C0 is 917 mm × 1297 mm, and D0 771 mm × 1090 mm. Series C is the basis for envelope formats.

This German standardisation work was accompanied by contributions from other countries, and the published DIN paper-format concept was soon introduced as a national standard in many other countries, for example, Belgium (1924), Netherlands (1925), Norway (1926), Switzerland (1929), Sweden (1930), Soviet Union (1934), Hungary (1938), Italy (1939), Finland (1942), Uruguay (1942), Argentina (1943), Brazil (1943), Spain (1947), Austria (1948), Romania (1949), Japan (1951), Denmark (1953), Czechoslovakia (1953), Israel (1954), Portugal (1954), Yugoslavia (1956), India (1957), Poland (1957), United Kingdom (1959), Venezuela (1962), New Zealand (1963), Iceland (1964), Mexico (1965), South Africa (1966), France (1967), Peru (1967), Turkey (1967), Chile (1968), Greece (1970), Zimbabwe (1970), Singapore (1970), Bangladesh (1972), Thailand (1973), Barbados (1973), Australia (1974), Ecuador (1974), Colombia (1975) and Kuwait (1975).

It finally became both an international standard (ISO 216) as well as the official United Nations document format in 1975 and it is today used in almost all countries on this planet, with the exception of North America, Peru, Colombia, and the Dominican Republic.

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.[4]

The main advantage of this system is its scaling. Rectangular paper with an aspect ratio of 2 has the unique property that, when cut or folded in half midway between its longer sides, each half has the same 2 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 2 side-by-side along their longer side, they form a larger rectangle with the aspect ratio of 2 and double the area of each individual sheet.

The ISO system of paper sizes exploits these properties of the 2 aspect ratio. In each series of sizes (for example, series A), the largest size is numbered 0 (for example, A0), and each successive size (for example, A1, A2, etc.) 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 (for example, an A4 sheet is folded in half to make a brochure with size A5 pages. 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 weight in grams (g) per area in square metres (abbreviated g/m2 or gsm).[5] One can derive the grammage of other sizes by arithmetic division in g/m2. A standard A4 sheet made from 80 g/m2 paper weighs 5 g, as it is 1/16 (four halvings, ignoring rounding) of an A0 page. Thus the weight, and the associated postage rate, can be easily approximated 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 ("untrimmed") paper sizes

## Properties of the 3 ISO-series

### A series

Paper in the A series format has an aspect ratio of 2 (≈ 1.414, when rounded). A0 is defined so that it has an area of 1 m2 before rounding to the nearest millimetre. Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the area of the preceding paper size and rounding down, so that the long side of A(n + 1) is the same length as the short side of An. Hence, each next size is nearly exactly half of the prior size. So, an A1 page can fit 2 A2 pages inside the same area.

The most used of this series is the size A4, which is 210 mm × 297 mm (8.27 in × 11.7 in) and thus almost exactly 0.0625 square metres (96.9 sq in) in area. For comparison, the letter paper size commonly used in North America (8 12 in × 11 in, 216 mm × 279 mm) is about 6 mm (0.24 in) wider and 18 mm (0.71 in) shorter than A4. Then, the size of A5 paper is half of A4, as 148 mm × 210 mm (5.8 in × 8.3 in).[6][7]

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, x/y, will be the same as that of a rectangle half its size, y/x/2, which means that x/y = y/x/2, which reduces to x/y = 2; in other words, an aspect ratio of 1:2.

Knowing the aspect ratio gives us the tools we need to calculate the first terms of the A-series paper's geometric sequence. Given Ln and Wn which serve as the geometric sequences for the length and width of paper An, we will know that:

${\displaystyle {\begin{cases}L_{0}\times W_{0}=1\\W_{0}\times {\sqrt {2}}=L_{0}\end{cases}}}$

Hence, in order to solve for W0 (the first term of Wn), we substitute L0 from the second equation of the system to find:

{\displaystyle {\begin{aligned}W_{0}\times {\sqrt {2}}\times W_{0}&=1\\\left(W_{0}\right)^{2}\times {\sqrt {2}}&=1\\\left(W_{0}\right)^{2}&={\frac {1}{\sqrt {2}}}\\W_{0}&={\sqrt {\frac {1}{\sqrt {2}}}}\\&=\left(2^{-{\frac {1}{2}}}\right)^{\frac {1}{2}}\\&=2^{-{\frac {1}{4}}}\\&={\frac {1}{\sqrt[{4}]{2}}}\end{aligned}}}

Now that we have the first term of Wn, we can find the first term of Ln through another substitution:

{\displaystyle {\begin{aligned}L_{0}\times {\frac {1}{\sqrt[{4}]{2}}}&=1\\{\frac {L_{0}}{1}}\times {\frac {1}{\sqrt[{4}]{2}}}&=1\\{\frac {\sqrt[{4}]{2}}{\sqrt[{4}]{2}}}&=1\\L_{0}&={\sqrt[{4}]{2}}\end{aligned}}}

Finally now we have the first terms of the geometric sequences Ln and Wn respectively, which are:

{\displaystyle {\begin{aligned}L_{0}&={\sqrt[{4}]{2}}\\W_{0}&={\frac {1}{\sqrt[{4}]{2}}}\end{aligned}}}

By finding the first terms of the geometric sequences for length and width, we are able to find the general algebraic equation to find the length and width of an A-series paper of any size n, simply by multiplying the expression for the first term with the common ratio ${\displaystyle {\frac {1}{\sqrt {2}}}}$. Note that the reason that the common ratio is the reciprocal of ${\displaystyle {\sqrt {2}}}$, because as n increases in An, the dimensions of the paper gets smaller (i.e. is divided) by a factor of ${\displaystyle {\sqrt {2}}}$.

Hence, the formula that gives the length (i.e. larger border) of the paper size An and prior to rounding is:

${\displaystyle L_{n}={\sqrt[{4}]{2}}\times \left({\frac {1}{\sqrt {2}}}\right)^{n}=2^{{\frac {1}{4}}-{\frac {2}{n}}}}$

Likewise, the formula that gives the width (i.e. smaller border) of the paper size An can be likewise found as the following geometric sequence:

${\displaystyle W_{n}={\frac {1}{\sqrt[{4}]{2}}}\times \left({\frac {1}{\sqrt {2}}}\right)^{n}=2^{-{\frac {1}{4}}-{\frac {n}{2}}}}$

Note that the width of an A-series paper of the size An is simply the length of an A-series paper of the previous size An-1. Hence, it is usually not necessary to explicitly express the formula to calculate the width. Rather, one can simply substitute for n-1 in the formula for length instead.

The paper size An thus has the dimension (before rounding):

${\displaystyle L_{n}\times W_{n}=L_{n}\times L_{n-1}}$

and area (before rounding)

2-n m2.

The above equations allow us to calculate a table of values for the parameters of A-series paper. Note that in practice, these values are not used - but rather they are rounded to the nearest millimetre. For the actual, practical dimensions of A-series paper, the chart in the previous section may be referred to.

Exact (algebraic) parameters for paper size of An
Paper Length (meters) Width (meters) Area (square meters)
A0 ${\displaystyle 2^{\frac {1}{4}}}$ ${\displaystyle 2^{-{\frac {1}{4}}}}$ ${\displaystyle 1}$
A1 ${\displaystyle 2^{-{\frac {1}{4}}}}$ ${\displaystyle 2^{-{\frac {3}{4}}}}$ ${\displaystyle {\frac {1}{2}}}$
A2 ${\displaystyle 2^{-{\frac {3}{4}}}}$ ${\displaystyle 2^{-{\frac {5}{4}}}}$ ${\displaystyle {\frac {1}{4}}}$
A3 ${\displaystyle 2^{-{\frac {5}{4}}}}$ ${\displaystyle 2^{-{\frac {7}{4}}}}$ ${\displaystyle {\frac {1}{8}}}$
A4 ${\displaystyle 2^{-{\frac {7}{4}}}}$ ${\displaystyle 2^{-{\frac {9}{4}}}}$ ${\displaystyle {\frac {1}{16}}}$
An ${\displaystyle 2^{{\frac {1}{4}}-{\frac {2}{n}}}}$ ${\displaystyle 2^{-{\frac {1}{4}}-{\frac {2}{n}}}}$ ${\displaystyle {\frac {1}{2^{n}}}}$

The length of the long side of An can be calculated as

lAn = ⌊1000 ⋅ 21 - 2n/4 + 0.2⌋ mm

(brackets represent the floor function).

As the aspect ratio of the A-series paper is 2, it is easy to scale (i.e. convert) between different A-series papers. Given an A-series paper of the size An, the scaling factor required to convert it to a size of Am is:

${\displaystyle S=\left({\sqrt {2}}\right)^{m-n}}$

Hence, in order to convert the dimensions of an A3 paper to the smaller A4, every dimension of the A3 paper must be multiplied by a factor of ${\displaystyle \left({\sqrt {2}}\right)^{3-4}\approx 0.7071}$.

### 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 2, and folding one in half (and rounding down to the nearest millimetre) gives the next in the series. The shorter side of B0 is exactly 1 metre.

The length of the long side of Bn can be calculated as:

lBn = ⌊1000 ⋅ 21 - n/2 + 0.2⌋ mm

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 (which is identical to the ISO A series).[8] Thus, the lengths of JIS B series paper are 1.5 ≈ 1.22 times those of A-series paper. By comparison, the lengths of ISO B series paper are 42 ≈ 1.19 times those of A-series paper.

### C series

The C series formats are geometric means between the B series and A series formats with the same number (e.g., C2 is the geometric mean between B2 and A2). The width to height ratio is 2 as in the A and B series. The C series formats are used mainly for envelopes. An unfolded A4 page will fit into a C4 envelope. C series envelopes follow the same ratio principle as the A series pages. For example, if an A4 page is folded in half so that it is A5 in size, it will fit into a C5 envelope (which will be the same size as a C4 envelope folded in half). The lengths of ISO C series paper are therefore 82 ≈ 1.09 times those of A-series paper.

A, B, and C paper fit together as part of a geometric progression, with ratio of successive side lengths of 82, though there is no size half-way between Bn and A(n − 1): A4, C4, B4, "D4", A3, ...; there is such a D-series in the Swedish extensions to the system.

The length of the long side of Cn can be calculated as:

lCn = ⌊1000 ⋅ 23 - 4n/8 + 0.2⌋ mm

## 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 1:2; 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 they 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 1:2 are popular in paper folding, such as origami, where they are sometimes called "A4 rectangles" or "silver rectangles".[9] In other contexts, the term "silver rectangle" can also refer to a rectangle in the proportion 1:(1 + 2), known as the silver ratio.

## Matching technical pen widths

Rotring Rapidographs in ISO nib sizes

An important adjunct to the ISO paper sizes, particularly the A series, are the technical drawing line widths specified in ISO 128, and the matching technical pen widths of 0.13, 0.18, 0.25, 0.35, 0.5, 0.7, 1.0, 1.40, and 2.0 mm, as specified in ISO 9175-1. Colour codes are assigned to each size to facilitate easy recognition by the drafter. These sizes increase by a factor of 2, so that particular pens can be used on particular sizes of paper, and then the next smaller or larger size can be used to continue the drawing after it has been reduced or enlarged, respectively. For example, a continuous thick line on A0 size paper shall be drawn with a 0.7 mm pen, the same line on A1 paper shall be drawn with a 0.5 mm pen, and finally on A2, A3, or A4 paper it shall be drawn with a 0.35 mm pen.[4][10][11]

 Linewidth (mm) Colour 0.10 0.13 0.18 0.25 0.35 0.50 0.70 1.0 1.4 2.0 Maroon Violet Red White Yellow Brown Blue Orange Green Gray
Micronorm logo

The earlier DIN 6775 standard upon which ISO 9175-1 is based also specified a term and symbol for easy identification of pens and drawing templates compatible with the standard, called Micronorm, which may still be found on some technical drafting equipment.

## References

1. "International Paper Sizes & Formats". Paper Sizes. Retrieved 29 June 2020.
2. Lichtenberg, Georg Christoph (February 7, 2006) [Written October 25, 1786]. "Lichtenberg's letter to Johann Beckmann" (in German and English). Translated by Kuhn, Markus. University of Cambridge. Retrieved May 10, 2016. Published in Lichtenberg, Georg Christoph (1990). Joost, Ulrich; Schöne, Albrecht (eds.). Briefwechsel [Correspondence] (in German). Volume III (1785–1792). Munich: Beck. pp. 274–75. ISBN 3-406-30958-5. Retrieved May 10, 2016.
3. Kuhn, Markus (October 8, 2005). "Loi sur le timbre (No. 2136)" [Law of Taxation (No. 2136)]. Retrieved May 11, 2016. Kuhn includes copies of pages from the journal article that announced the law: Republic of France (November 3, 1798). "Loi sur le timbre (Nº 2136)". Bulletin des lois de la République (in French). Paris (237): 1–2.
4. ^ a b Kuhn, Markus. "International standard paper sizes". Retrieved 30 August 2017.
5. International Organization for Standardization (2012). "ISO 536:2012(en): Paper and board — Determination of grammage". ISO Browsing Platform (3 ed.). §  3.1 note 1. Missing or empty |url= (help)
6. "A Paper Sizes – A0, A1, A2, A3, A4, A5, A6, A7, A8, A9". papersizes.org. Retrieved 2018-08-02.
7. "International Paper Sizes, Dimensions, Format & Standards". PaperSize. Retrieved 2018-10-05.
8. "Japanese B Series Paper Size". Retrieved 2010-04-18.
9. Lister, David. "The A4 rectangle". The Lister List. England: British Origami Society. Retrieved 2009-05-06.
10. "Technical drawing pen sizes". Designing Buildings Wiki. Retrieved 30 August 2017.
11. Bell, Steven. "Pen Sizes and Line Types". Metrication.com. Retrieved 30 August 2017.