Editing Paper size (section)

===A series<span class="anchor" id="ISO A"></span><span class="anchor" id="DIN A"></span><span class="anchor" id="SIS A"></span><span class="anchor" id="JIS A"></span>===
[[File:A size illustration2.svg|thumb|right|A size chart illustrating the ISO A series]]

There are 11 sizes in the A series, designated A0–A10, all of which have an aspect ratio of <math>\frac{a}{b} = \sqrt{2} \approx 1.41421\ldots</math>, where ''a'' is the long side and ''b'' is the short side.

Since A series sizes share the same aspect ratio <math>(\sqrt{2}),</math> they can be scaled to other A series sizes without being distorted, and two sheets can be reduced to fit on exactly one sheet without any cutoff or margins.

The A0 base size is defined as having an area of 1&nbsp;m{{sup|2}}; given an aspect ratio of <math>\sqrt{2}</math>, the dimensions of A0 are:

<math>\sqrt[4]2\,\mathrm{m}</math> by <math>\frac1\sqrt[4]2\,\mathrm{m}</math>.

or, rounded to the nearest millimetre, {{cvt|1189|x|841|mm|inch}}.

A series sizes are related in that the smaller dimension of a given size is the larger dimension of the next smaller size, and folding an A series sheet in half in its larger dimension—that is, folding it in half parallel to its short edge—results in two halves that are each the size of the next smaller A series size. As such, a folded brochure of a given A-series size can be made by folding sheets of the next larger size in half, e.g. A4 sheets can be folded to make an A5 brochure. The fact that halving a sheet with an aspect ratio of <math>\sqrt{2}</math> results in two sheets that themselves both have an aspect ratio of <math>\sqrt{2}</math> is proven as follows:

<math>\frac{a}{b} = \sqrt{2},</math>

where ''a'' is the long side and ''b'' is the short side. The aspect ratio for the new dimensions of the folded paper is:

<math>\frac{b}{\frac{a}{2}} = 2\frac{b}{a} = 2\frac{1}{\sqrt{2}} = \sqrt{2} = \frac{a}{b}.</math>

The advantages of basing a paper size upon an aspect ratio of <math>\sqrt{2}</math> were noted in 1786 by the German scientist and philosopher [[Georg Christoph Lichtenberg]].<ref>{{cite web |url=http://www.cl.cam.ac.uk/~mgk25/lichtenberg-letter.html |title=Lichtenberg's letter to Johann Beckmann |publisher=Markus Kuhn |date=7 February 2006 |access-date=3 April 2023 |archive-url=https://web.archive.org/web/20111231214454/http://www.cl.cam.ac.uk/~mgk25/lichtenberg-letter.html |archive-date=31 December 2011 |url-status=live}}</ref> He also observed that some raw sizes already adhered to that ratio so that when a sheet is folded, the length to width ratio does not change.

Briefly after the introduction of the [[metric system]], a handful of new paper formats equivalent to modern ones were developed in France, having been proposed by the mathematician [[Lazare Carnot]], and published for judicial purposes in 1798 during the [[French Revolution]]:<ref name="B237">{{cite journal |title=Loi sur le timbre (Nº 2136) |trans-title=Stamp Act (No. 2136) |date=3 November 1798 |journal=Bulletin des Lois de la République |issue=237 |pages=1–2 |language=fr |publisher=Republic of France |location=Paris |url=http://www.cl.cam.ac.uk/~mgk25/loi-timbre.html |url-status=live |access-date=2024-01-20 |archive-url=https://web.archive.org/web/20090426170239/http://www.cl.cam.ac.uk/~mgk25/loi-timbre.html |archive-date=26 April 2009 |via=Markus Kuhn }}</ref>

* {{lang|fr|Grand registre}} (A2)
* {{lang|fr|Moyen papier}} (A3)
* {{lang|fr|Grand papier}} (B3)
* {{lang|fr|Petit papier}} (B4)
* {{lang|fr|Demi feuille}} (B5)
* {{lang|fr|Effets de commerce}} (B6)

These were never widely adopted, however.

Early in the 20th century, the ratio was used to specify the [[#Weltformat|''world format'']] starting with {{cvt|1|cm}} as the short edge of the smallest size. Walter Porstmann started with the largest sizes instead, assigning one an area of {{cvt|1|m2}} (A0) and the other a short edge of {{cvt|1|m}} (B0). He thereby turned the forgotten French sizes (relatively few in number) into a logically-simple and comprehensive plan for a full range of paper sizes, while introducing systematic alphanumeric monikers for them. Generalized to nothing less than four series, this system was introduced as a [[#German original|DIN standard (DIN 476)]] in Germany in 1922, replacing a vast variety of other paper formats. Even today, the paper sizes are called "DIN A4" ({{IPA|de|diːn.ʔaː.fiːɐ̯|IPA}}) in everyday use in Germany and Austria.

The DIN 476 standard spread quickly to other countries. Before the outbreak of [[World War II]], it had been adopted by the following countries in Europe:

{{div col|colwidth=22em}}
* [[Belgium]] (1924)
* [[Netherlands]] (1925)
* [[Norway]] (1926)
* [[Finland]] (1927)
* [[Switzerland]] (1929)
* Sweden (1930) with [[#Swedish extension|later extensions]]
* [[Soviet Union]] (1934) with [[#Soviet variants|custom extensions]]
* [[Hungary]] (1938)
* Italy (1939)
{{div col end}}

During World War II, the standard spread to South America and was adopted by [[Uruguay]] (1942), [[Argentina]] (1943) and Brazil (1943), and afterwards spread to other countries:

{{div col|colwidth=22em}}
* [[Australia]] (1974)
* [[Austria]] (1948)
* [[Bangladesh]] (1972)
* [[Barbados]] (1973)
* [[Chile]] (1968)
* [[Colombia]] (1975)
* [[Czechoslovakia]] (1953)
* [[Denmark]] (1953)
* [[Ecuador]] (1974)
* [[France]] (1967)
* [[Greece]] (1970)
* [[Iceland]] (1964)
* [[India]] (1957) with [[#Indian variants|custom extensions]]
* [[Iran]] (1948)
* [[Ireland]] (1959)
* [[Israel]] (1954)
* [[Japan]] (1951) with [[#Japanese variation|different B series]]
* [[Kuwait]] (1975)
* [[Mexico]] (1965)
* [[New Zealand]] (1963)
* [[Peru]] (1967)
* [[Poland]] (1957)
* [[Portugal]] (1954)
* [[Rhodesia]] (1970)
* [[Romania]] (1949)
* [[Singapore]] (1970)
* [[South Africa]] (1966)
* [[Spain]] (1947)
* [[Thailand]] (1973)
* [[Turkey]] (1967)
* [[United Kingdom]] (1971)<ref>{{cite web|author=Metrication Board|title=Final report of the Metrication Board|date=1980|publisher=Department of Trade and Industry Consumer and Competition Policy Directorate|page=17|url=http://ukma.org.uk/sites/default/files/met1980.pdf|archive-url=https://web.archive.org/web/20130501034405/http://ukma.org.uk/sites/default/files/met1980.pdf|access-date=29 September 2021|archive-date=1 May 2013}}</ref>
* [[Venezuela]] (1962)
* [[SFRY|Yugoslavia]](1956)
{{div col end}}

By 1975, so many countries were using the German system that it was established as an [[International Organization for Standardization|ISO]] standard, as well as the official United Nations document format. By 1977, A4 was the standard letter format in 88 of 148 countries. Today the standard has been adopted by all countries in the world except the United States and Canada. In Mexico, [[Costa Rica]], [[Colombia]], [[Venezuela]], Chile, and the [[Philippines]], the [[Letter (paper size)|US letter]] format is still in common use, despite their official adoption of the ISO standard.

The weight of an A-series sheet of a given [[Paper density#Basis weight|paper weight]] can be calculated by knowing the ratio of its size to the A0 sheet. For example, an A4 sheet is {{frac|16}} the size of an A0 sheet, so if it is made from {{nowrap|80 g/m{{sup|2}}}} paper, it weighs {{frac|16}} of {{cvt|80|g}}, which is {{cvt|5|g}}.