The ratio that makes it work
ISO 216 is built on one property: a rectangle with sides in the ratio 1 : √2 keeps that exact ratio when folded or cut in half along its longer side. Fold such a rectangle in half and the two halves are smaller rectangles in the same 1:√2 proportion as the original, just rotated 90 degrees. No other ratio does this: fold a square in half and you get two rectangles that are not squares; fold a typical US Letter sheet in half and the halves are a noticeably different shape from the original.
That single property is why an A4 sheet folded in half becomes two A5 sheets rather than two oddly proportioned scraps, and why a document formatted for A4 scales cleanly onto A3 at exactly double the area with nothing cropped or distorted.
Where the sequence starts
The series starts at A0, defined to have an area of exactly one square metre while keeping the 1:√2 ratio, which works out to 841 × 1189 mm. Every size after that is the previous one folded in half along the long edge:
| Size | Dimensions |
|---|---|
| A0 | 841 × 1189mm |
| A1 | 594 × 841mm |
| A2 | 420 × 594mm |
| A3 | 297 × 420mm |
| A4 | 210 × 297mm |
| A5 | 148 × 210mm |
| A6 | 105 × 148mm |
A4, four folds down from A0, is one sixteenth of a square metre in area, which is where its familiar 210 × 297mm comes from: not a round number chosen on purpose, but the arithmetic result of halving a one-square-metre sheet four times while preserving the ratio.
B and C: the two series most people have never heard named
The A series leaves noticeable gaps for some uses, so ISO 216 defines a B series alongside it, sized so each B number sits at the geometric mean between the matching A number and the one above it. B5 is common for hardback books; B0 and similar large B sizes turn up in posters. A related standard, ISO 269, defines a C series specifically for envelopes: a C4 envelope is sized to hold an unfolded A4 sheet exactly, a C5 holds an A4 sheet folded once in half, and a C6 holds it folded into quarters, which is why office envelope sizes are not arbitrary either.
Where the idea came from
The 1:√2 folding property was described as early as 1786, in a letter from the German physicist and satirist Georg Christoph Lichtenberg, who noted the geometric curiosity without it becoming a standard. It stayed a curiosity for well over a century. Germany's DIN 476 standard formalised it into an actual paper size system in 1922, and that German standard is the direct ancestor of A4: the dimensions have not changed since. ISO adopted the same system internationally as ISO 216 in 1975, and most of the world outside North America uses it today.
Checking a sheet against the standard
Because the ratio is exact, a genuine A4 sheet should measure 210mm on the short edge and 297mm on the long edge to within the manufacturer's cutting tolerance, usually well under a millimetre. The Paper Size Guide shows every standard size drawn to your screen's calibrated scale, so a printed sheet can be laid against the display and checked directly, the same way a bank card checks a screen's calibration in the first place.