Editing ISO 216 (section)

===A series===

Paper in the A series format has an aspect ratio of {{math|{{sqrt|2}}}} (≈ 1.414, when rounded). A0 is defined so that it has an area of {{cvt|1|m2|ft2|lk=on}} before rounding to the nearest {{convert|1|mm}}. Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the area of the preceding paper size and rounding down, so that the long side of {{Nowrap|A(''n'' + 1)}} is the same length as the short side of A''n''. Hence, each next size is nearly exactly half the area of the prior size. So, an A1 page can fit two A2 pages inside the same area.

The most used of this series is the size A4, which is {{cvt|210|x|297|mm|sigfig=3}} and thus almost exactly {{convert|1/16|m2|m2 sqin|4}} in area. For comparison, the [[Letter (paper size)|letter]] paper size commonly used in North America ({{convert|8+1/2|x|11|in|sigfig=3|abbr=on|disp=semicolon}}) is about {{Nowrap|6 mm}} ({{Nowrap|0.24 in}}) wider and {{Nowrap|18 mm}} ({{Nowrap|0.71 in}}) shorter than A4. Then, the size of A5 paper is half of A4, i.e. {{Nowrap|148 mm}} × {{Nowrap|210 mm}} ({{Nowrap|5.8 in}} × {{Nowrap|8.3 in}}).<ref>{{cite web |title= A Paper Sizes – A0, A1, A2, A3, A4, A5, A6, A7, A8, A9 |url= https://www.papersizes.org/a-paper-sizes.htm |website= papersizes.org |access-date= 2 August 2018 }}</ref><ref>{{cite web |title= International Paper Sizes, Dimensions, Format & Standards |url= https://papersize.co/ |website= PaperSize |access-date= 5 October 2018 }}</ref>

The geometric rationale for using the [[square root of 2]] is to maintain the aspect ratio of each subsequent rectangle after cutting or folding an A-series sheet in half, perpendicular to the larger side. Given a rectangle with a longer side, ''x'', and a shorter side, ''y'', ensuring that its aspect ratio, {{sfrac|''x''|''y''}}, will be the same as that of a rectangle half its size, {{sfrac|''y''|''x''/2}}, which means that {{math|1={{sfrac|''x''|''y''}} = {{sfrac|''y''|''x''/2}}}}, which reduces to {{math|1={{sfrac|''x''|''y''}} = {{sqrt|2}}}}; in other words, an aspect ratio of {{math|1:{{sqrt|2}}}}.

Any {{math|A''n''}} paper can be defined as {{math|1=A''n'' = ''S'' × ''L''}}, where (measuring in metres)

:<math>\text{A}_n = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{n + \frac{1}{2}}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{n - \frac{1}{2}}
\end{cases}</math>

Therefore

:<math>\text{A0} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{0 + \frac{1}{2}} \approx 0.841\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{0 - \frac{1}{2}} \approx 1.189\,\text{m}
\end{cases}</math>, {{pad}} <math>\text{A1} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{1 + \frac{1}{2}} \approx 0.595\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{1 - \frac{1}{2}} \approx 0.841\,\text{m}
\end{cases}</math> {{pad}} <math>\text{A2} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{2 + \frac{1}{2}} \approx 0.420\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{2 - \frac{1}{2}} \approx 0.595\,\text{m}
\end{cases}</math> {{pad}} Etc.