International standard paper sizes

Standard paper sizes like ISO A4 are widely used all over the world today. This text explains the ISO 216 paper size system and the ideas behind its design.

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.

A diagram demonstrating the sqrt(2) 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:

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.

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 following table shows the width and height of all ISO A and B paper formats, as well as the ISO C envelope formats. The dimensions are in millimeters:

A Series Formats B Series Formats C Series Formats
4A0 1682 × 2378
2A0 1189 × 1682
A0 841 × 1189 B0 1000 × 1414 C0 917 × 1297
A1 594 × 841 B1 707 × 1000 C1 648 × 917
A2 420 × 594 B2 500 × 707 C2 458 × 648
A3 297 × 420 B3 353 × 500 C3 324 × 458
A4 210 × 297 B4 250 × 353 C4 229 × 324
A5 148 × 210 B5 176 × 250 C5 162 × 229
A6 105 × 148 B6 125 × 176 C6 114 × 162
A7 74 × 105 B7 88 × 125 C7 81 × 114
A8 52 × 74 B8 62 × 88 C8 57 × 81
A9 37 × 52 B9 44 × 62 C9 40 × 57
A10 26 × 37 B10 31 × 44 C10 28 × 40

The allowed tolerances are ±1.5 mm for dimensions up to 150 mm, ±2 mm for dimensions above 150 mm up to 600 mm, and ±3 mm for dimensions above 600 mm. Some national equivalents of ISO 216 specify tighter tolerances, for instance DIN 476 requires ±1 mm, ±1.5 mm, and ±2 mm respectively for the same ranges of dimensions.

Application examples

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

The main advantage of the ISO standard paper sizes becomes obvious for users of copying machines:

Example 1:

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:

fromto A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
A0 100% 71% 50% 35% 25% 18% 12.5% 8.8% 6.2% 4.4% 3.1%
A1 141% 100% 71% 50% 35% 25% 18% 12.5% 8.8% 6.2% 4.4%
A2 200% 141% 100% 71% 50% 35% 25% 18% 12.5% 8.8% 6.2%
A3 283% 200% 141% 100% 71% 50% 35% 25% 18% 12.5% 8.8%
A4 400% 283% 200% 141% 100% 71% 50% 35% 25% 18% 12.5%
A5 566% 400% 283% 200% 141% 100% 71% 50% 35% 25% 18%
A6 800% 566% 400% 283% 200% 141% 100% 71% 50% 35% 25%
A7 1131% 800% 566% 400% 283% 200% 141% 100% 71% 50% 35%
A8 1600% 1131% 800% 566% 400% 283% 200% 141% 100% 71% 50%
A9 2263% 1600% 1131% 800% 566% 400% 283% 200% 141% 100% 71%
A10 3200% 2263% 1600% 1131% 800% 566% 400% 283% 200% 141% 100%

Not only the operation of copying machines in offices and libraries, but also repro photography, microfilming, and printing are simplified by the 1:sqrt(2) aspect ratio of ISO paper sizes.

Example 2:

If you prepare a letter, you will have to know the weight of the content in order to determine the postal fee. This can be very conveniently calculated with the ISO A series paper sizes. Usual typewriter and laser printer paper weighs 80 g/m². An A0 page has an area of 1 m², and the next smaller A series page has half of this area. Therefore, the A4 format has an area of 1/16 m² and weighs with the common paper quality 5 g per page. If we estimate 20 g for a C4 envelope (including some safety margin), then you will be able to put 16 A4 pages into a letter before you reach the 100 g limit for the next higher postal fee.

Calculation of the mass of books, newspapers, or packed paper is equally trivial. You probably will not need such calculations often, but they nicely show the beauty of the concept of metric paper sizes.

Using standard paper sizes saves money and makes life simpler in many applications. For example, if all scientific journals used only ISO formats, then libraries would have to buy only very few different sizes for the binders. Shelves can be designed such that standard formats will fit in exactly without too much wasted shelf volume. The ISO formats are used for surprisingly many things besides office paper: the German citizen ID card has format A7, both the European Union and the U.S. (!) passport have format B7, and library microfiches have format A6. In some countries (e.g., Germany) even many brands of toilet paper have format A6.

Further details

Calculating the dimensions

Although the ISO paper sizes are specified in the standard with the width and height given in millimeters, the dimensions can also be calculated with the following formulas:

Format Width [m] Height [m]
An 2−1/4−n/2 21/4−n/2
Bn 2n/2 21/2−n/2
Cn 2−1/8−n/2 23/8−n/2

The actual millimeter dimensions in the standard have been calculated by progressively rounding down any division-by-two result, as the small program iso-paper.c demonstrates. This guarantees that two A(n+1) pages together are never larger than an An page.

Aspect ratios other than sqrt(2)

Sometimes, paper formats with a different aspect ratio are required for labels, tickets, and other purposes. These should preferably be derived by cutting standard series sizes into 3, 4, or 8 equal parts, parallel with the shorter side, such that the ratio between the longer and shorter side is greater than the square root of two. Some example long formats in millimeters are:

1/3 A4 99 × 210
1/4 A4 74 × 210
1/8 A4 37 × 210
1/4 A3 105 × 297
1/3 A5 70 × 148

The 1/3 A4 format (99 × 210 mm) is also commonly applied for reduced letterheads for short notes that contain not much more than a one sentence message and fit without folding into a DL envelope.

Envelope formats

For postal purposes, ISO 269 and DIN 678 define the following envelope formats:

Format Size [mm] Content Format
C6 114 × 162 A4 folded twice = A6
DL 110 × 220 A4 folded twice = 1/3 A4
C6/C5 114 × 229 A4 folded twice = 1/3 A4
C5 162 × 229 A4 folded once = A5
C4 229 × 324 A4
C3 324 × 458 A3
B6 125 × 176 C6 envelope
B5 176 × 250 C5 envelope
B4 250 × 353 C4 envelope
E4 280 × 400 B4

The DL format is the most widely used business letter format. DL probably originally stood for “DIN lang”, but ISO 269 now explains this abbreviation instead more diplomatically as “Dimension Lengthwise”. Its size falls somewhat out of the system and equipment manufacturers have complained that it is slightly too small for reliable automatic enveloping. Therefore, DIN 678 introduced the C6/C5 format as an alternative for the DL envelope.

Window envelopes, A4 letterheads, folding marks and standard layouts

There exists no international standard yet for window envelopes and matching letterhead layouts. There are various incompatible national standards, for example:

According to ISO 11180 and Universal Postal Union standards, an international postal address should be not longer than 6 lines with up to 30 characters each. This requires a maximum area of 76.2 × 38.1 mm with the commonly used typewriter character width of 2.54 mm (1/10") and a baseline distance of 6.35 mm (1/4").

The Universal Postal Union Letter Post Regulations specify a standard position of the address on the envelope, which is within 140 mm from the right edge, at least 40 mm from the top edge, and is surrounded by at least 15 mm unprinted envelope to the left, right and below of the address text.

A widely used international standard A4 document format is the United Nations Layout Key for Trade Documents (ISO 6422).

Folding larger pages to A4 for filing

DIN 824 describes a method of folding A0, A1, etc. pages to A4 format for filing. This clever technique ensures that there remains a 20 mm single-layer margin for filing holes, that the page can be unfolded and folded again without being removed from the file, and that the label field at the bottom-left corner of technical drawings ends up in correct orientation on top of the folded page in the file.

Folder and file sizes

ISO 623 specifies the sizes of folders and files intended to receive either A4 sheets or simple folders (without back) that are not designed for any particular filing system or cabinet. The sizes specified are those of the overall rectangular surface when the folders or files are folded, exclusive any margin or tabs. Simple folders without back or mechanism are 220 × 315 mm large. Folders and files with a very small back (less than 25 mm) with or without mechanism are 240 × 320 mm large. Files with wide back (exceeding 25 mm) are 250 × 320 mm (without a mechanism) or 290 × 320 mm if they include a mechanism. All these are maximum dimensions. Standardizing folder and file sizes helps to optimize shelf designs and provides a uniform look and handling even if folders from various manufacturers are used.

Filing holes

ISO 838 specifies that, for filing purposes, two holes of 6±0.5 mm diameter can be punched into the sheets. The centers of the two holes are 80±0.5 mm apart and have a distance of 12±1 mm to the nearest edge of the sheet. The holes are located symmetrically in relation to the axis of the sheet or document. Any format that is at least as large as A7 can be filed using this system.

Not specified in ISO 838, but also widely used, is an upwards compatible 4-hole system. Its two middle holes correspond to ISO 838, plus there are two additional holes located 80 mm above and below these to provide for more stability. This way, sheets with four punched holes can also be filed in ISO 838 2-hole binders. This system is also known under the nickname "888", presumably because the three gaps between the holes are all 8 cm wide. Some hole punches have on their paper guide not only markings for "A4", "A5", and "A6", but also for "888". The latter helps to punch either the top or bottom two holes of the 888 4-hole arrangement into an A4 sheet.

Technical drawing pen sizes

Technical drawing pens follow the same size-ratio principle. The standard sizes differ by a factor sqrt(2): 2.00 mm, 1.40 mm, 1.00 mm, 0.70 mm, 0.50 mm, 0.35 mm, 0.25 mm, 0.18 mm, 0.13 mm. So after drawing with a 0.35 mm pen on A3 paper and reducing it to A4, you can continue with the 0.25 mm pen. (ISO 9175-1)

Ruled writing paper

There seems to be no international standard yet for ruled writing paper. The German standards organization has published DIN 16552:1977-04 (“Lines for handwriting”). That system is widely used, at least in Germany, by primary school teachers to specify which school exercise books pupils should use at which stage of learning how to write. Writing paper with fine gray 5 mm grid lines seems to be very popular in many countries.

Untrimmed paper formats

All A and B series formats described so far are trimmed paper end sizes, i.e. these are the dimensions of the paper delivered to the user or reader. Other ISO standards define the format series RA and SRA for untrimmed raw paper, where SRA stands for “supplementary raw format A” (“sekundäres Rohformat A”). These formats are only slightly larger than the corresponding A series formats. Sheets in these formats will be cut to the end format after binding. The ISO RA0 format has an area of 1.05 m² and the ISO SRA0 format has an area of 1.15 m². These formats also follow the sqrt(2)-ratio and half-area rule, but the dimensions of the start format have been rounded to the full centimeter. The common untrimmed paper formats that printers order from the paper manufacturers are

RA Series Formats SRA Series Formats
RA0 860 × 1220 SRA0 900 × 1280
RA1 610 × 860 SRA1 640 × 900
RA2 430 × 610 SRA2 450 × 640
RA3 305 × 430 SRA3 320 × 450
RA4 215 × 305 SRA4 225 × 320

The RA and SRA dimensions are also used as roll widths in rotating printing presses.

Overhead projectors

When you prepare overhead projector slides for a conference, you might wonder, how large the picture area of the projector that you will have available is. ISO 7943-1 specifies two standard sizes of overhead projector picture areas: Type A is 250 × 250 mm (corners rounded with a radius less than 60 mm) and Type B is 285 × 285 mm (corners rounded with a radius less than 40 mm or cut off diagonally no more than 40 mm). Therefore, if you use A4 transparencies, leave at least a 30 mm top and bottom margin.

Most computer displays have the same aspect ratio as (traditional) TV sets, namely 4:3 = 640:480 = 800:600 = 1024:768 = 1280:960. If you prepare presentation slides, I recommend that you arrange your layout inside a 280 × 210 mm field and make sure that you leave at least 20 mm margin on the left and right side. This way, you plan for the aspect ratio of a TV/VGA projector and ensure at the same time that you can print on A4 transparencies such that every standard overhead projector will show all parts of your slides.

Identification cards

ISO 7810 specifies three formats for identification cards:

ID-1 is the common format for banking cards (0.76 mm thick) and is also widely used for business cards and driver’s licences. Some people prefer A8 (74 × 52 mm) for business cards. The standard passport format is B7 (= ID-3), the German ID card has A7 (= ID-2) format and the European Union driver’s licence is an ID-1 card.

History of the ISO paper formats

One of the oldest written records regarding the sqrt(2) aspect ratio for paper sizes is a letter that the physics professor Georg Christoph Lichtenberg (University of Göttingen, Germany, 1742-1799) wrote 1786-10-25 to Johann Beckmann. In it, Lichtenberg explains the practical and aesthetic advantages of the sqrt(2) aspect ratio, and of his discovery that paper with that aspect ratio was commonly available at the time. (There are also suggestions that the task to find a paper format that is similar to itself after being cut in half appeared as a question in mathematics exams as early as 1755.)

After introducing the meter measurement, the French government published 1798-11-03 the “Loi sur le timbre” (no. 2136), a law on the taxation of paper that defined several formats that already correspond exactly to the modern ISO paper sizes: “Grand registre” = ISO A2, “grand papier” = ISO B3, “moyen papier” = ISO A3, “petit papier” = ISO B4, “demi feuille” = ISO B5, “effets de commerce” = ISO 1/2 B5.

The French format series never became widely known and was quickly forgotten again. The A, B, and C series paper formats, which are based on the exact same design principles, were completely independently reinvented over a hundred years after the “Loi sur le timbre” in Germany by Dr. Walter Porstmann. They were adopted as the German standard DIN 476 in 1922 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. (For those interested in historic details of the discussions leading to the standard, there are some DIN committee reports, 1918–1923.)

Porstmann’s DIN paper-format concept was convincing, and 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), 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), Simbabwe (1970), Singapur (1970), Bangladesh (1972), Thailand (1973), Barbados (1973), Australia (1974), Ecuador (1974), Columbia (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, leaving North America as the only remaining exception. 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 then. [Source: Helbig/Hennig 1988]

Note: The Lichtenberg Ratio – used by the standard paper format series – is occasionally confused with the Golden Ratio (which Euclid referred to as the “extreme and mean ratio”). The Lichtenberg Ratio is defined by the equation a/b = 2b/a = sqrt(2), whereas the Golden Ratio is defined by a/b = (a+b)/a = b/(a−b) = (1 + sqrt(5))/2. While aesthetically pleasing properties have been attributed to both, the Lichtenberg Ratio has the advantage of preserving the aspect ratio when cutting a page into two. The Golden Ratio, on the other hand, preserves the aspect ratio when cutting a maximal square from the paper, a property that seems not particularly useful for office applications. The Golden Ratio was for a while a more fashionable topic in the antique and renaissance arts literature and it has a close connection to the Fibonacci sequence in mathematics.


References

This text summarizes and explains the content of the following international standards:

The following standards contain related information but are not covered here completely:

These standards are available from:

International Organization for Standardization
Case postale 56
1, rue de Varembé
CH-1211 Genève 20
Switzerland

phone: +41 22 749 01 11
fax: +41 22 733 34 30
web: www.iso.org

The most comprehensive source of information about the ISO and North American paper formats and many related standards, as well as their respective histories, is the book

DIN also produced a brief German prospectus with information about the history of the DIN paper sizes:


Created 1996-10-29 – last modified 2006-05-02 – http://www.cl.cam.ac.uk/~mgk25/iso-paper.html