ISO 269 sizes
(mm × mm)

C Series
C0	917 × 1297
C1	648 × 917
C2	458 × 648
C3	324 × 458
C4	229 × 324
C5	162 × 229
C6	114 × 162
C7/6	81 × 162
C7	81 × 114
C8	57 × 81
C9	40 × 57
C10	28 × 40
DLE	110 × 220

ISO 216 sizes
(mm × mm)

B Series
B0	1000 × 1414
B1	707 × 1000
B2	500 × 707
B3	353 × 500
B4	250 × 353
B5	176 × 250
B6	125 × 176
B7	88 × 125
B8	62 × 88
B9	44 × 62
B10	31 × 44

ISO 216 sizes
(mm × mm)

A Series
A0	841 × 1189
A1	594 × 841
A2	420 × 594
A3	297 × 420
A4	210 × 297
A5	148 × 210
A6	105 × 148
A7	74 × 105
A8	52 × 74
A9	37 × 52
A10	26 × 37

ISO 216 specifies international standard

International Organization for Standardization

The International Organization for Standardization , widely known as ISO, is an international standard-setting body composed of representatives from various national standards organizations. Founded on February 23, 1947, the organization promulgates worldwide proprietary, industrial and commercial...

 (ISO) paper size

Paper size

Many paper size standards conventions have existed at different times and in different countries. Today there is one widespread international ISO standard and a localised standard used in North America . The paper sizes affect writing paper, stationery, cards, and some printed documents...

s used in most countries in the world today. It defines the "A" and "B" series of paper sizes, including A4, the most commonly available size. Two supplementary standards, ISO 217

ISO 217

The ISO 217:1995 standard defines the RA and SRA paper formats.These paper series are untrimmed raw paper. RA stands for “raw format A” and SRA stands for “supplementary raw format A”. The RA and SRA formats are slightly larger than the corresponding A series formats. This allows bleed on printed...

 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 have the same aspect ratio

Aspect ratio

The aspect ratio of a shape is the ratio of its longer dimension to its shorter dimension. It may be applied to two characteristic dimensions of a three-dimensional shape, such as the ratio of the longest and shortest axis, or for symmetrical objects that are described by just two measurements,...

, . This ratio has the unique property that when cut or folded in half lengthwise, the halves also have the same aspect ratio. Each ISO paper size is one half of the area of the next size up.

History

The advantages of basing a paper size upon an aspect ratio of √2 were already noted in 1786 by the German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 scientist Georg Christoph Lichtenberg

Georg Christoph Lichtenberg

Georg Christoph Lichtenberg was a German scientist, satirist and Anglophile. As a scientist, he was the first to hold a professorship explicitly dedicated to experimental physics in Germany...

, in a letter to Johann Beckmann. The formats that became A2, A3, B3, B4 and B5 were developed in France, and published in 1798 during the French Revolution

French Revolution

The French Revolution , sometimes distinguished as the 'Great French Revolution' , was a period of radical social and political upheaval in France and Europe. The absolute monarchy that had ruled France for centuries collapsed in three years...

, but were subsequently forgotten.
Early in the twentieth century, Dr Walter Porstmann turned Lichtenberg's idea into a proper system of different paper sizes. Porstmann's system was introduced as a DIN standard

Deutsches Institut für Normung

is the German national organization for standardization and is that country's ISO member body. DIN is a Registered German Association headquartered in Berlin...

 (DIN 476) in Germany

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 in 1922, replacing a vast variety of other paper formats. Even today the paper sizes are called "DIN Ax" in everyday use in Germany and Austria.

The main advantage of this system is its scaling: if a sheet with an aspect ratio of √2 is divided into two equal halves parallel to its shortest sides, then the halves will again have an aspect ratio of √2. Folded brochures of any size can be made by using sheets of the next larger size, e.g. A4 sheets are folded to make A5 brochures. The system allows scaling without compromising the aspect ratio from one size to another – as provided by office photocopiers, e.g. enlarging A4 to A3 or reducing A3 to A4. Similarly, two sheets of A4 can be scaled down and fit exactly 1 sheet without any cutoff or margins.

The weight of each sheet is also easy to calculate given the basis weight in grams

Metric system

The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...

 per square metre (g/m² or "gsm"). Since an A0 sheet has an area of 1 m², its weight in grams is the same as its basis weight in g/m². A standard A4 sheet made from 80 g/m² paper weighs 5g, as it is one 16th (four halvings) of an A0 page. Thus the weight, and the associated postage rate, can be easily calculated by counting the number of sheets used.

ISO 216 and its related standards were first published between 1975 and 1995:

• ISO 216:1975, defining the A and B series of paper sizes
• ISO 269:1985, defining the C series for envelopes
• ISO 217:1995, defining the RA and RSA series of raw ("untrimmed") paper sizes

The A series

Paper in the A series format has a aspect ratio, although this is rounded to the nearest millimetre. A0 is defined so that it has an area of 1 square metre

Square metre

The square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...

, prior to the above mentioned rounding. Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the preceding paper size, cutting parallel to its shorter side (so that the long side of A(n+1) is the same length as the short side of An, again prior to rounding).

The most frequently used of this series is the size A4 which is 210 millimetre. For comparison, the letter

Letter (paper size)

Letter or US Letter is the most common paper size for office use in several countries, including the United States, Canada, Mexico, Bolivia, Colombia, Venezuela, the Philippines, and Chile. It measures 8.5 by 11 inches ....

 paper size commonly used in North America

North America

North America is a continent wholly within the Northern Hemisphere and almost wholly within the Western Hemisphere. It is also considered a northern subcontinent of the Americas...

 (8.5" x 11") is approximately 6 mm (0.24 in) wider and 18 mm (0.71 in) shorter than A4.

The geometric rationale behind the square root of 2

Square root of 2

The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...

 is to maintain the aspect ratio of each subsequent rectangle after cutting the sheet in half, perpendicular to the larger side. Given a rectangle with a longer side, x, and a shorter side, y, the following equation shows how the aspect ratio of a rectangle compares to that of a half rectangle: which reduces to or an aspect ratio of .

The formula that gives the larger border of the paper size A in metres and without rounding off is the geometric sequence: . The paper size A thus has the dimension × .

The exact millimetre measurement of the long side of A is given by .

The B series

The B series are 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 means that each step in size: B0, A0, B1, A1, B2 … is smaller than the previous by an equal scaling. In a similar manner to the A series; the lengths of the B series still have the ratio , and folding one in half gives the next in the series. The shorter side of B0 is exactly 1m.

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). Thus, the lengths of JIS B series paper are times those of A-series paper. By comparison, the lengths of ISO B series paper are times those of A-series paper.

For the ISO B series, the exact millimetre measurement of the long side of B is given by .

The 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 C series formats are used mainly for envelope

Envelope

An envelope is a common packaging item, usually made of thin flat material. It is designed to contain a flat object, such as a letter or card....

s. An 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).

A, B, and C paper fit together as part of a geometric progression

Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

, with ratio of successive side lengths of 2^1/8, though there is no size half-way between Bn and An-1: A4, C4, B4, "D4", A3, …; there is such a D-series in the Swedish extensions to the system.

The exact millimetre measurement of the long side of C is given by .

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.

A, B, C comparison

ISO/DIN paper sizes in millimetres and in inch

Inch

An inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...

es

	A Series Formats		B Series Formats		C Series Formats
size	mm	inches	mm	inches	mm	inches
0	841 × 1189	33.1 × 46.8	1000 × 1414	39.4 × 55.7	917 × 1297	36.1 × 51.1
1	594 × 841	23.4 × 33.1	707 × 1000	27.8 × 39.4	648 × 917	25.5 × 36.1
2	420 × 594	16.5 × 23.4	500 × 707	19.7 × 27.8	458 × 648	18.0 × 25.5
3	297 × 420	11.7 × 16.5	353 × 500	13.9 × 19.7	324 × 458	12.8 × 18.0
4	210 × 297	8.3 × 11.7	250 × 353	9.8 × 13.9	229 × 324	9.0 × 12.8
5	148 × 210	5.8 × 8.3	176 × 250	6.9 × 9.8	162 × 229	6.4 × 9.0
6	105 × 148	4.1 × 5.8	125 × 176	4.9 × 6.9	114 × 162	4.5 × 6.4
7	74 × 105	2.9 × 4.1	88 × 125	3.5 × 4.9	81 × 114	3.2 × 4.5
8	52 × 74	2.0 × 2.9	62 × 88	2.4 × 3.5	57 × 81	2.2 × 3.2
9	37 × 52	1.5 × 2.0	44 × 62	1.7 × 2.4	40 × 57	1.6 × 2.2
10	26 × 37	1.0 × 1.5	31 × 44	1.2 × 1.7	28 × 40	1.1 × 1.6

Application

Before the adoption of ISO 216, many different paper formats were used internationally. These formats did not fit into a coherent system and were defined in terms of non-metric units.

The ISO 216 formats are organized around the ratio ; 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

United States

The United States of America is a federal constitutional republic comprising fifty states and a federal district...

, Canada

Canada

Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...

 and Venezuela

Venezuela

Venezuela , officially called the Bolivarian Republic of Venezuela , is a tropical country on the northern coast of South America. It borders Colombia to the west, Guyana to the east, and Brazil to the south...

, which use the Letter, Legal and Executive system. Although they have also officially adopted the ISO 216 paper format, Mexico

Mexico

The United Mexican States , commonly known as Mexico , is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of...

, The Philippines and Chile

Chile

Chile ,officially the Republic of Chile , is a country in South America occupying a long, narrow coastal strip between the Andes mountains to the east and the Pacific Ocean to the west. It borders Peru to the north, Bolivia to the northeast, Argentina to the east, and the Drake Passage in the far...

 also use most U.S. paper sizes in ordinary usage.

Rectangular

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

 sheets of paper with the ratio are popular in paper folding

Paper folding

Paper craft is the collection of art forms employing paper or card as the primary artistic medium for the creation of three-dimensional objects. It is the most widely used material in arts and crafts. It lends itself to a wide range of techniques, it can for instance be folded, cut, glued,...

, where they are sometimes called "A4 rectangles" or "silver rectangles". However, in other contexts, the term "silver rectangle" can also refer to a rectangle in the proportion , known as the silver ratio

Silver ratio

In mathematics, two quantities are in the silver ratio if the ratio between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller. This defines the silver ratio as an irrational mathematical constant,...

.

External links

• International standard paper sizes: ISO 216 details and rationale
• ISO 216 at iso.org