The practical and aesthetic advantages of the sqrt(2) aspect ratio for paper sizes were probably first noted by the physics professor Georg Christoph Lichtenberg (University of Göttingen, Germany, 1742-1799) in a letter that he wrote 1786-10-25 to Johann Beckmann. After introducing the meter measurement, the French government published 1794-11-03 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.

The DIN paper formats were soon also introduced in many other countries, for example, Belgium (1924), Switzerland (1929), Soviet Union (1934), Japan (1951), India (1957), United Kingdom (1959), Mexico (1965), France (1967), Australia (1974), Columbia and Kuwait (1975). Porstmann's DIN paper format system 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.

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.

In the ISO paper size system, the height-to-width ratio of all pages is the square root of two (1.4142 : 1). 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 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 following table shows the width and height of all ISO A and B paper formats, as well as the ISO C envelope formats.

Due to popular demand, I have prepared an unofficial table with the ISO sizes in inch fractions. Each listed inch fraction has the smallest denominator that keeps the value within the ISO 216 tolerance limits. Product designers should use the official millimeter values instead.

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.