# 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 underlying principle is that when rectangles with width/length ratio $1:\sqrt{2}$ are cut or folded in half the rectangles thus formed retain the original width/length ratio.

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, but were later forgotten.

The aspect ratio used by this standard was mentioned in a letter by the German Georg Christoph Lichtenberg, written on 25 October 1786.

• 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

Paper in the A series format has a $1:\sqrt{2} \approx 0.707$ aspect ratio, although this is rounded to the nearest millimetre. 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 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 (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 geometric rationale behind the square root of 2 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: $\ x/y = y/(x/2)$ which reduces to $x/y = \sqrt{2}$ or an aspect ratio of $1 : \sqrt{2}$

The formula that gives the larger border of the paper size An in metres and without rounding off is the geometric sequence: an = 21 / 4 − n / 2. The paper size An thus has the dimension an × an + 1.

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