# 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.

ISO paper sizes are based on the metric system. The square-root-of-two ratio does not allow the height and width of pages to be given simple whole metric sizes. The sizes therefore, are rounded to whole millimetre lengths. As the weight of paper in the metric system is specified in grams per square metre (gsm), a simple method of calculating the mass or weight of a publication is possible, where the size and number of pages is 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 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.

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:

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 or pre-sets for the following frequently used magnification or reduction factors:

*Example 2:*

When preparing a letter, knowing the weight of the paper helps when calculating the cost of mailings. This can be very conveniently calculated with the ISO A series paper sizes. Standard stationery paper weighs 80 gsm. An A0 page has an area of 1 sq m, and the next smaller A series page has half of this area. Therefore the A4 format has an area of 1/16 sq m and weighs 5 grams per page. Allowing an estimate of 20 grams for a C4 envelope, 16 A4 pages can be included before the initial 100 gram limit is reached.

The ISO formats are used for surprisingly many things besides office paper: the German citizen ID card has format A7, and the European Union passport uses the B7 format, library microfiches use the A6 format.

**Aspect ratios**

There are occasions when paper sizes other than the standard A series are required, such as labels, tickets etc. These can be achieved by cutting standard series sizes into 3, 4, or 8 equal parts, parallel to the shorter side, in order that the ratio between the longer and shorter side is greater than the square root of two. Some example long formats in millimetres are:

The 1/3 A4 format (99 x 210 mm) is also commonly used for reduced letterheads or compliment slips, or for short notes that contain not much more than a single 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**

**C6**

**DL**

**C6/C5**

**C5**

**C4**

**C3**

**B6**

**B5**

**B4**

**E4**

**Size (mm)**

The DL format is the most widely used business letter format. Its size falls somewhat out of the system and equipment manufacturers have complained that it is slightly too small for the reliable automatic enveloping, therefore DIN 678 introduced the C6/C5 format as an alternative for DL.

There is no current ISO standard for envelopes with an address window, there is however a corresponding DIN standard. DIN 680 specifies that a transparent address window should measure 90mm x 45 mm with its left side 20 mm from the left edge of the envelope. For C6, DL, and C6/C5 envelopes, the bottom of the window should be 15 mm from the bottom edge of the envelope. For C4 envelopes, the top of the window should be either 27 or 45 mm from the top edge of the envelope.

**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 end user. The ISO 217:1995 standard defines the RA (raw format A) and SRA (supplementary raw format A) formats for untrimmed raw paper. These formats are only slightly larger than the corresponding A formats. Sheets in these formats are cut to the final size format after printing and binding. The ISO RA0 format has an area of 1.05 sq m and the ISO SRA0 format has an area of 1.15 sq m. These also follow the square root of 2 ratio and the half-area rule, but the dimensions of the base format have been rounded to the full centimetre. The common untrimmed paper formats printers order from the paper manufacturers or merchants are:

**RA Series Formats**

**RA0**

**RA1**

**RA2**

**RA3**

**RA4**

**SRA Series Formats**

**SRA0**

**SRA1**

**SRA2**

**SRA3**

**SRA4**

*This page is based upon the full text; International Standard Paper Sizes by Markus Kuhn.Markus Kuhn is a Senior Lecturer at the University of Cambridge Computer Laboratory and a fellow of Wolfson College.*