A series Paper in the A series format has an
aspect ratio of (≈ 1.414, when rounded). A0 is defined so that it has an area of 1 m2 (about 11 ft2) before rounding to the nearest . 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 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 two A2 pages (in
landscape orientation) fit together over an A1 page (in
portrait orientation), an A3 page is half an A2 page, A4 is half an A3 and so on. The most used of this series is the
A4 paper size, which is and thus almost exactly in area. For comparison, the
letter paper size commonly used in North America () is about () wider and () shorter than A4. Then, the size of A5 paper is half of A4, i.e. × ( × ). 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, , will be the same as that of a rectangle half its size, , which means that , which reduces to ; in other words, an aspect ratio of . Any paper can be defined as , where (measuring in metres) \text{A}_n = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{n + 1/2}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{n - 1/2} \end{cases} Therefore \text{A}_0 = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{0 + 1/2} \approx 0.841\,\text{m}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{0 - 1/2} \approx 1.189\,\text{m} \end{cases} \text{A}_1 = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{1 + 1/2} \approx 0.595\,\text{m}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{1 - 1/2} \approx 0.841\,\text{m} \end{cases} \text{A}_2 = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{2 + 1/2} \approx 0.420\,\text{m}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{2 - 1/2} \approx 0.595\,\text{m} \end{cases} etc.
B series The B series is defined in the standard as follows: "A subsidiary series of sizes is obtained by placing the
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 , 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
JIS defines to have 1.5 times the area of the corresponding JIS A series (which is identical to the ISO A series). Thus, the lengths of JIS B series paper are ≈ 1.22 times those of A-series paper. By comparison, the lengths of ISO B series paper are ≈ 1.19 times those of A-series paper (and ≈ 1.41 times the area). Any paper (according to the ISO standard) can be defined as , where (measuring in metres) \text{B}_n = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^n\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{n - 1} \end{cases} Therefore \text{B}_0 = \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} \text{B}_1 = \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} \text{B}_2 = \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} 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 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 , though there is no size half-way between B
n and : A4, C4, B4, "D4", A3, ...; there is such a D-series in the
Swedish extensions to the system. The lengths of ISO C series paper are therefore ≈ 1.09 times those of A-series paper. The C series formats are used mainly for
envelopes. 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 paper can be defined as , where (measuring in metres) \text{C}_n = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{n + 1/4}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{n - 3/4} \end{cases} Therefore \text{C}_0 = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{0 + 1/4} \approx 0.917\,\text{m}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{0 - 3/4} \approx 1.297\,\text{m} \end{cases} \text{C}_1 = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{1 + 1/4} \approx 0.648\,\text{m}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{1 - 3/4} \approx 0.917\,\text{m} \end{cases} \text{C}_2 = \begin{cases} S = \left(\sqrt{\frac{1}{2}}\,\right)^{2 + 1/4} \approx 0.458\,\text{m}\\ L = \left(\sqrt{\frac{1}{2}}\,\right)^{2 - 3/4} \approx 0.648\,\text{m} \end{cases} etc. ==Tolerances==