The key to understanding the most widely used international sizes of paper is aspect — the ratio of the longer edge of a sheet to the shorter. All sheets in this system are rectangular, and all feature the same aspect. This means that an image designed to fit on one international sheet can be enlarged or reduced to fit another using the same percentage of size change in both the horizontal and vertical directions. In other words, no "stretching" is necessary to prevent nonproportional margins.
In numerical terms, the aspect is 1.414, or more precisely the square root of two. Although this number might seem inconvenient, it was chosen so that if a sheet is cut in half parallel to the shorter edges, each of the resulting sheets also has an aspect of 1.414. On the other hand, folding or cutting parallel to a longer edge does not preserve the ratio.
Moreover, if a sheet is folded in half with the crease parallel to the shorter edges, each page thereby formed has that same aspect of 1.414; this latter fact is of interest to printers of books, as printing is often done on large sheets which are then folded in half several times and then cut to the final size. No other number has this property.
Perhaps surprisingly, the sizes are not in alphabetical order, the sequence being A C B D:
|⇐ smaller||larger ⇒|
Be aware that some unrelated systems use similar-looking designators for sheets whose sizes are a round number of inches, unrelated to the square root of two.
The table below lists many of the possible paper sizes. Some of these are recognized by the International Organization for Standardization (ISO) in their standards numbered 216 (A and B series) and 269 (C series). Additionally, various national standards bodies recognize some of the paper sizes that the ISO omits.
Except in the ISO column, the following numbers are rounded to four significant figures. Greater precision is of little use because practically all papers are made from cellulose, and consequently expand as they absorb moisture from the air, and contract as they release moisture. The most common rounding practice, as adopted by the ISO, is to discard any fraction of a millimeter. For instance, 840.9 mm becomes 840 mm and 324.2 mm becomes 324 mm.
Manufacturers do not come close to offering all these sizes. The only ones seen with any frequency are:
One inch equals 25.4 millimeters.
Consecutive sizes, when stacked, look like this:
Here are links to five such diagrams ready to print at 100% scale:
|to be printed on this size of sheet …|
largest size depicted …
smallest size depicted …
|or make your own …||source code|
Some computer monitors have aliasing problems with these diagrams, but printers generally do better.
For reference are shown some powers of two calculated to six places:
|2||+5/8 = 1.542211||2||0/8 = 1.000000|
|2||+4/8 = 1.414214||2||−1/8 = 0.917004|
|2||+3/8 = 1.296840||2||−2/8 = 0.840896|
|2||+2/8 = 1.189207||2||−3/8 = 0.771105|
|2||+1/8 = 1.090508||2||−4/8 = 0.707107|
Here are the general formulas for the sheet sizes, where r = 2+1/8 and s = 2−4/8:
|Larger sizes||Smaller sizes|
|mD0||r+5 × √m||r+1 × √m||Dn||r+5 × sn||r+1 × sn|
|mB0||r+4 × √m||r 0 × √m||Bn||r+4 × sn||r 0 × sn|
|mC0||r+3 × √m||r−1 × √m||Cn||r+3 × sn||r−1 × sn|
|mA0||r+2 × √m||r−2 × √m||An||r+2 × sn||r−2 × sn|
|m should be a power of two, such as 2, 4, 8, 16, 32 …||n should be a nonnegative integer, such as 0, 1, 2, 3, 4 …|
Were non-integers allowed in designations, A2.5, C2.75, B3, and D3.25 would all indicate the same size of sheet.
A Swedish standard (SIS 014711) introduces a finer granularity with the E, F and G (but not H) series. Here are formulas for the smaller sizes, with the obvious extension to larger:
|omitted||r+5.5 × sn||r+1.5 × sn|
|Dn||r+5.0 × sn||r+1.0 × sn|
|Fn||r+4.5 × sn||r+0.5 × sn|
|Bn||r+4.0 × sn||r 0.0 × sn|
|Gn||r+3.5 × sn||r−0.5 × sn|
|Cn||r+3.0 × sn||r−1.0 × sn|
|En||r+2.5 × sn||r−1.5 × sn|
|An||r+2.0 × sn||r−2.0 × sn|
|n should be a nonnegative integer, such as 0, 1, 2, 3, 4 …|
The nonalphabeticality of the sequence is patent.
For some applications, a sheet of paper needs to be square. One example is the label for the end of a box containing a cylindrical item, while another case involves the paper-folding art of origami where the norm is the square sheet. Because if there is a standard it is obscure, a suggestion is presented here.
With the rectangular sheets discussed above, the ratio of the longer edge to the shorter is the square root of two, which is closely approximated by the ratio 7 ÷ 5 = 1.4; the difference is just a shade over one percent. This error is small enough to present no major problem, because with paper sizes a certain amount of variation is inevitable anyway; one major reason for this follows.
A square piece of paper may not remain square when the weather changes. This is because typical manufacturing equipment carries the liquid that will become paper on a conveyor belt. In this fluid environment, the cellulose fibers that will be the main ingredient of the finished product tend to align themselves parallel to the direction of travel -- this is called the machine direction. Perpendicular to that is the cross direction. After manufacturing is complete, there is a risk that under variations of humidity, the percent of expansion or contraction in the machine direction will differ from that in the cross direction. For this reason along with the usual errors of measurement that can occur in any industrial process, the attainment of absolute squareness is difficult if not impossible.
However, there are two approaches to generating sheets that are nearly enough square to be useful. Under either method, the first step is thus: in one large rectangular sheet, make four evenly-spaced cuts (yielding five strips) perpendicular to the shorter edge. For instance, a B0 sheet (1414 by 1000 mm) would be cut into five strips measuring 1414 by 200 mm. Then choose one of the following procedures:
In major production work, the squares alternatively might be cut from a roll of paper several meters wide and hundreds of meters long, not from standard sheets. In that case there is every reason to cut to the precise square size, as waste can assuredly be minimized.
These square sizes need names. In an arbitrary choice, S (for square) is selected, with the three succeeding letters of the Latin alphabet. The chart below gives a few examples of square sizes, and the rectangular sheets from which they are efficiently cut.
|Source Sheet||Resultant Sheet|
|D0||1542. × 1091.||V2||218.2 × 218.2
||B0||1414. × 1000.||T0||200.0 × 200.0
||C0||1297. × 917.0||U0||183.4 × 183.4
||A0||1189. × 840.9||S0||168.2 × 168.2
||D1||1091. × 771.1||V1||154.2 × 154.2
||B1||1000. × 707.1||T1||141.4 × 141.4
||C1||917.0 × 648.4||U1||129.7 × 129.7
||A1||840.9 × 594.6||S1||118.9 × 118.9
||each source sheet yields 35 resultant sheets