Iso_216

ISO 216 specifies international standard (ISO) paper sizes used in most countries in the world today. It is the standard which defines the commonly available A4 paper size.

The international ISO standard is based on the German DIN standard 476 (DIN 476) from 1922. Some of the formats contained therein were independently invented in France during its revolution and later forgotten. [1] The aspect ratio used by this norm was mentioned in a letter by the German Georg Christoph Lichtenberg written 1786-10-25. [2]

• ISO 216:1975, defines two series of paper sizes: A and B
• ISO 269:1985, defines a C series for envelopes
• ISO 217:1995, defines two untrimmed series of raw paper sizes: RA and SRA

A series

A size chart illustrating the ISO A series.

Paper in the A series format has a $1:\backslash sqrt\{2\}$ aspect ratio, although this is rounded to the nearest millimeter. A0 is defined so that it has an area of 1 m², 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 An+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 (210 × 297 mm). A4 paper is 6 mm narrower and 18 mm longer than the "Letter" paper size, 8½ × 11 inches (216 × 279 mm), commonly used in North America.

The geometrical rationale behind the square root of 2 being the aspect ratio is that a half of a rectangle in which this is the proportion between sides is a smaller rectangle having the same proportion between sides. This guarantees that for example, at the same time two A4 sheet placed side-by-side have the same size of an A3 sheet, while the A4 and A3 sheets have exactly the same ratio between long and short sides.

The formula that gives the larger border of the paper size A$n$ in metres and without rounding off is the geometric sequence: $a\_n\; =\; 2^\{1/4\; -\; n/2\}$. The paper size A$n$ thus has the dimension $a\_n$ × $a\_\{n+1\}$.

The exact millimeter measurement of the long side of A$n$ is given by $\backslash left\; \backslash lfloor\; 1000/(2^\{(2n-1)/4\})+0.2\; \backslash right\; \backslash rfloor$.

B series

A size chart illustrating the ISO B series.

The B series formats are geometric means between the A series format with the same number and the A series format with one lower number. For example, B1 is a geometric mean between A1 and A0. The sides of B0 are 1 m to $\backslash sqrt\{2\}$ m.

There is also an incompatible Japanese B series defined by the JIS. The lengths of JIS B series paper are approximately 1.22 times those of A-series paper.

The exact millimeter measurement of the long side of B$n$ is given by $\backslash left\; \backslash lfloor\; 1000/(2^\{(n-1)/2\})+0.2\; \backslash right\; \backslash rfloor$.

C series

The C series formats are geometric means between the B series format with the same number and the A series format with the same number, (e.g., C2 is the geometric mean between B2 and A2). The C series formats are used mainly for envelopes. 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).

The exact millimeter measurement of the long side of C$n$ is given by $\backslash left\; \backslash lfloor\; 1000/(2^\{(4n-3)/8\})+0.2\; \backslash right\; \backslash rfloor$.

Tolerances

The tolerances specified in the standard are:

• ±1.5 mm for dimensions up to 150 mm,
• ±2.0 mm for lengths in the range 150 to 600 mm, and
• ±3.0 mm for any dimension above 600 mm.

A, B, C comparison

ISO/DIN paper sizes in millimetres and in inches

	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 $1:\backslash sqrt\{2\}$; two sheets next to each other together have the same ratio, sideways. This simplifies copying (e.g. two A4 sheets in reduced size onto one A4 sheet; an A4 sheet in magnified size on an A3 sheet; or half an A4 sheet in magnified size on an A4 sheet). The principal countries not generally using the ISO paper sizes are the United States and Canada, which use the Letter, Legal and Executive system. (Canada uses a P-series of sizes, which are the US paper sizes rounded to metric dimensions.)

Rectangular sheets of paper with the ratio $1:\backslash sqrt\{2\}$ are popular in paper folding, where they are sometimes called "A4 rectangles" or "silver rectangles".[3] (Confusingly, "silver rectangle" can also refer to a rectangle in the proportion $1:(1+\backslash sqrt\{2\})$), known as the silver ratio.)

See also

• Paper size
• Letter (paper size)

External links

• International standard paper sizes: ISO 216 details and rationale
• A4 paper artwork
• ISO 216 at iso.org

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