Editing ISO 216 (section)

==Properties==
===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.

===B series===
The B series is defined in the standard as follows: "A subsidiary series of sizes is obtained by placing the [[Geometric mean|geometrical means]] between adjacent sizes of the A series in sequence." The use of the geometric mean makes each step in size: B0, A0, B1, A1, B2 ... smaller than the previous one by the same factor. As with the A series, the lengths of the B series have the ratio {{math|{{sqrt|2}}}}, and folding one in half (and rounding down to the nearest millimetre) gives the next in the series. The shorter side of B0 is exactly 1 metre.

There is also an incompatible Japanese B series which the [[Japanese Industrial Standard|JIS]] defines to have 1.5 times the area of the corresponding JIS A series (which is identical to the ISO A series).<ref>{{cite web|url=http://www.paper-sizes.com/uncommon-paper-sizes/japanese-b-series-paper-size|title=Japanese B Series Paper Size|access-date=18 April 2010}}</ref> Thus, the lengths of JIS B series paper are {{math|{{sqrt|1.5}}}} ≈ 1.22 times those of A-series paper. By comparison, the lengths of ISO B series paper are {{math|{{sqrt|2|4}}}} ≈ 1.19 times those of A-series paper.

Any {{math|B''n''}} paper (according to the ISO standard) can be defined as {{math|1=B''n'' = ''S'' × ''L''}}, where (measuring in metres)

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

Therefore

:<math>\text{B0} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{0} = 1\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{0 - 1} \approx 1.414\,\text{m}
\end{cases}</math>, {{pad}} <math>\text{B1} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{1} \approx 0.707\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{1 - 1} = 1\,\text{m}
\end{cases}</math> {{pad}} <math>\text{B2} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{2} = 0.5\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{2 - 1} \approx 0.707\,\text{m}
\end{cases}</math> {{pad}} Etc.

===C series===
The C series formats are geometric means between the B series and A series formats with the same number (e.g. C2 is the geometric mean between B2 and A2). The width to height ratio of C series formats is {{math|{{sqrt|2}}}} as in the A and B series. A, B, and C series of paper fit together as part of a [[geometric progression]], with ratio of successive side lengths of {{math|{{sqrt|2|8}}}}, though there is no size half-way between B''n'' and {{Nowrap|A(''n'' − 1)}}: A4, C4, B4, "D4", A3, ...; there is such a D-series in the [[Paper size#Swedish extensions|Swedish extensions]] to the system. The lengths of ISO C series paper are therefore {{math|{{sqrt|2|8}}}} ≈ 1.09 times those of A-series paper.

The C series formats are used mainly for [[envelope]]s. An unfolded A4 page will fit into a C4 envelope. Due to same width to height ratio, 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).

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

:<math>\text{C}_n = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{n + \frac{1}{4}}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{n - \frac{3}{4}}
\end{cases}</math>

Therefore

:<math>\text{C0} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{0 + \frac{1}{4}} \approx 0.917\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{0 - \frac{3}{4}} \approx 1.297\,\text{m}
\end{cases}</math>, {{pad}} <math>\text{C1} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{1 + \frac{1}{4}} \approx 0.648\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{1 - \frac{3}{4}} \approx 0.917\,\text{m}
\end{cases}</math> {{pad}} <math>\text{C2} = \begin{cases}
S = \left(\sqrt{\frac{1}{2}}\right)^{2 + \frac{1}{4}} \approx 0.458\,\text{m}\\
L = \left(\sqrt{\frac{1}{2}}\right)^{2 - \frac{3}{4}} \approx 0.648\,\text{m}
\end{cases}</math> {{pad}} Etc.