A ratio that survives being cut in half
Fold an A4 sheet in half across its short edge and the result is an A5 sheet: same shape, half the area. Fold it again and it's A6, still the same shape. That self-similarity is not a coincidence of the specific numbers 210 and 297 — it's the defining property of the whole ISO 216 A-series, and it only works for one particular aspect ratio.
The derivation: why √2 is the only ratio that works
Let a sheet have short side a and long side b, with aspect ratio r = b/a. Cut it in half across the long side: each half has dimensions a (unchanged) and b/2. For the half-sheet to be the same shape as the original, its long side divided by its short side must again equal r. Since a > b/2 for any ratio under 2, the half-sheet's long side is now a and its short side is b/2, giving a new ratio of a ÷ (b/2) = 2a/b. Setting that equal to the original ratio:
2a/b = b/a ⇒ 2a² = b² ⇒ b/a = √2
No other ratio reproduces itself under halving. √2 ≈ 1.41421 is the unique fixed point of this operation, which is why the entire A series — A0 down to A10 — is one shape at every size, rather than a new shape having to be chosen at each step.
A0: fixed by area, not by a memorized length
ISO 216 defines A0 by area, not by dimensions: exactly 1 m², at the √2 ratio. Solving a × b = 1 with b = a√2 gives a²√2 = 1, so a = 2−1/4 ≈ 0.840896 m and b = 21/4 ≈ 1.189207 m. Rounded to the nearest millimetre, A0 is 841 × 1189 mm. Every smaller size in the series is generated by repeatedly halving the long side and rounding down to the millimetre, which is why A4's true area, 210 × 297 mm = 62,370 mm² (0.06237 m²), is very slightly under the "exact" 1/16 m² (0.0625 m²) that four successive exact halvings of 1 m² would give: rounding to a whole millimetre at each step accumulates a tiny, deliberate loss.
| Size | Dimensions | Area |
|---|---|---|
| A0 | 841 × 1189 mm | 1.000 m² |
| A1 | 594 × 841 mm | 0.500 m² |
| A2 | 420 × 594 mm | 0.249 m² |
| A3 | 297 × 420 mm | 0.125 m² |
| A4 | 210 × 297 mm | 0.0624 m² |
| A5 | 148 × 210 mm | 0.0311 m² |
From a 1786 letter to a 1975 standard
The √2 property was noted at least as early as 1786, in a letter from the German physicist Georg Christoph Lichtenberg describing exactly this self-preserving ratio. It became a paper standard through Walter Porstmann, working at the Berlin standards institute, whose proposal became the German standard DIN 476 in 1922 — the direct ancestor of the international ISO 216 standard, first published in 1975. Most of the world uses ISO 216 sizes today; North America is the major exception.
US Letter: a shape with no such derivation
US Letter is 8.5 × 11 in, which is 215.9 × 279.4 mm. Its ratio is 11 ÷ 8.5 = 1.2941, nowhere near √2 (1.41421). There is no equivalent clean derivation behind 8.5 × 11: the dimensions descend from the size of hand papermaking moulds in use in the American colonial and early federal period, rather than from any formula, and the size was only formally locked in as the standard for US government documents in 1980, replacing a slightly different "Government Letter" format that had been in parallel use. Because Letter's ratio isn't the self-preserving one, folding it in half does not reproduce its own shape: half of an 8.5 × 11 in sheet is 5.5 × 8.5 in, a ratio of 8.5 ÷ 5.5 = 1.545 — a third, different shape again, not a scaled-down Letter.
B and C series: the same idea, shifted
ISO 216 also defines a B series and a related C series, both built on the identical √2 logic. Each B size is the geometric mean of the matching A size and the next larger one (B1 sits between A1 and A0, and so on), which makes the B series useful for posters, passports and other items that need a size between two A sizes rather than one that matches an A sheet exactly. The C series, defined in ISO 269, follows the same rule to produce envelope sizes: a C4 envelope (229 × 324 mm) is sized to hold an unfolded A4 sheet with a small margin, and a C5 envelope holds an A4 sheet folded once, in half. All three series share one aspect ratio precisely because that ratio is the only one that lets a sheet, an envelope and a folded insert all stay proportionally related to each other.
The printing consequence
A4 is 8.268 × 11.693 in; Letter is 8.5 × 11 in. The two are close enough to look interchangeable and different enough to cause a real problem: Letter is wider (8.5 in vs 8.268 in) but shorter (11 in vs 11.693 in). Printing an A4 document at 100% scale onto Letter paper crops roughly 0.693 in off the bottom of the page; printing a Letter document onto A4 leaves roughly 0.693 in of blank margin at the bottom instead. Every "print to fit" or "scale to page" option in office software exists largely to paper over this exact mismatch, because unlike the A-series, where every size is a scaled copy of every other, A4 and Letter are not scaled copies of each other at all — they are two different shapes that happen to be close to the same size.