# paper size

Many paper size standards conventions have existed at different times and in different countries. Today there is one widespread international ISO standard (including A4, B3, C4, etc.) and a local standard used in North America (including letter, legal, ledger, etc.). The paper sizes affect writing paper, stationery, cards, and some printed documents. The standards also have related sizes for envelopes.

### A series

The international paper size standard, ISO 216, is based on the German DIN 476 standard for paper sizes. ISO paper sizes are all based on a single aspect ratio of square root of 2, or approximately 1:1.4142. The base A0 size of paper is defined to have an area of 1 m2. Rounded to millimetres, the A0 paper size is 841 by 1,189 millimetres (33.1 in × 46.8 in).

Successive paper sizes in the series A1, A2, A3, and so forth, are defined by halving the preceding paper size across the larger dimension. The most frequently used paper size is A4 measuring 210 by 297 millimetres (8.3 in × 11.7 in).

The significant advantage of this system is its scaling: if a sheet with an aspect ratio of $\sqrt{2}$ is divided into two equal halves parallel to its shortest sides, then the halves will again have an aspect ratio of $\sqrt{2}$. 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 behavior of the aspect ratio is easily proven: Let $a$ and $b$ be the long side and the short side of the paper respectively. The imposed initial condition is that $a/b=\sqrt{2}$, let $c$be the length of the short side after folding it in half. That is $c=a/2$, if we take the ratio of the newly folded paper we have that:

$\frac{b}{c}=\frac{b}{\frac{a}{2}}=\frac{2}{\frac{a}{b}}=\frac{2}{\sqrt{2}}=\sqrt{2}$

Therefore the aspect ratio is preserved for the new dimensions of the folded paper.

Weights are easy to calculate as well: a standard A4 sheet made from 80 g/m2 paper weighs 5 g (as it is one 16th of an A0 page, measuring 1 m2), allowing one to easily compute the weight—and associated postage rate—by counting the number of sheets used.

The advantages of basing a paper size upon an aspect ratio of $\sqrt{2}$ were first noted in 1786 by the German scientist and philosopher Georg Christoph Lichtenberg.[2] Early in the 20th 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 (DIN 476) inGermany in 1922, replacing a vast variety of other paper formats. Even today the paper sizes are called “DIN A4” (IPA: [diːn.ʔaː.fiːɐ̯]) in everyday use in Germany and Austria. The term Lichtenberg ratio has recently been proposed for this paper aspect ratio.

According to some theorists, ISO 216 sizes are generally too tall and narrow for book production (see: Canons of page construction). European book publishers typically use metricated traditional page sizes for book production