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 heighttowidth 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 squarerootoftwo 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 squarerootoftwo 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 halfway between two numbers by addition, whereas the geometric mean is halfway between two numbers by multiplication. }
^{By the way: The Japanese JIS P 013861 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 }

^{2−n/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 divisionbytwo result, as the small program }^{isopaper.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.}