A4 paper is 210 by 297 millimeters. Those numbers have the unmistakable look of a committee compromise, perhaps reached after everyone with round-number ideas left for lunch.
They are not arbitrary. They are what remains when a rectangle is asked to do two jobs at once: keep the same shape every time it is cut in half, and begin with a sheet whose area is one square meter.
The first requirement determines the shape. The second determines the size. Together they produce the entire A-series family, from poster-sized A0 to A4 office paper and the smaller sheets beyond it.
The rectangle that survives halving
Call the short side of a sheet x and the long side y. Cut across the sheet parallel to its short side, splitting the long side in half. Each new sheet measures x by y/2, and after rotating it, the new long side is x.
For the smaller rectangle to have the same proportions as the original, the ratios must match:
original: y / x
half: x / (y / 2)
same shape means:
y / x = 2x / y
y² = 2x²
y / x = √2There is only one answer. The long side must be the short side multiplied by the square root of two, about 1.4142. The ISO paper-size system uses that ratio, within the small errors caused by rounding dimensions to whole millimeters.
Cut a perfect √2 rectangle in half and the two pieces are perfect √2 rectangles. Put two together along their long edges and the larger rectangle has the same shape too. A page can grow or shrink through the series without becoming squatter or skinnier.
One square meter, halved four times
A ratio does not tell a paper mill how large to make the first sheet. The A series gets its scale from area: A0 is defined to cover one square meter.
A one-square-meter rectangle with a √2 aspect ratio has theoretical sides of about 840.896 and 1,189.207 millimeters. Round those to practical dimensions and A0 becomes 841 by 1,189 millimeters. Its actual rounded area is 0.999949 square meters, close enough that arguing over the missing 51 square millimeters would be an excellent way to avoid doing any work.
Each step halves the previous sheet:
A0 841 × 1189 mm
A1 594 × 841 mm
A2 420 × 594 mm
A3 297 × 420 mm
A4 210 × 297 mm
A5 148 × 210 mmThe halves do not always land on whole millimeters. Half of 297 is 148.5, while standard A5 is 148 by 210 millimeters. The published dimensions are rounded down as the series progresses so that two smaller sheets never exceed the larger one. This is why the ratios wobble slightly around √2 instead of matching it digit for digit.
A shape designed for copying
The geometry becomes useful the moment a document changes size. Reduce an A3 page to A4 at 70.71 percent in each direction and the layout fits without changing proportions. Enlarge A4 to A3 at 141.42 percent and the same thing happens in reverse. Two A4 pages placed side by side make the shape of A3, so a copier can shrink a two-page spread onto one A4 sheet without inventing lopsided margins.
U.S. Letter paper does not have this property. Letter is 8.5 by 11 inches, a ratio of about 1.294. Fold it in half and rotate the result, and the ratio becomes about 1.545. The child is shaped differently from the parent.
The area rule supplies another small convenience. A4 is one-sixteenth of A0, so an A4 sheet has roughly one-sixteenth of a square meter of paper. At 80 grams per square meter, a common office weight, one sheet weighs almost exactly five grams. Paper dimensions, copier buttons, envelope sizes, and postage arithmetic all descend from the same decision.
The idea waited for a standard
The √2 trick predates ISO. In an October 1786 letter, German physicist Georg Christoph Lichtenberg described giving a student the problem of finding a sheet whose successive folio, quarto, and octavo divisions would all be similar. After solving it, he discovered that the writing paper in front of him already had the ratio. He wondered whether papermakers followed a rule or merely inherited the shape through tradition.
The institutional answer arrived much later. A history published by Germany's standards body credits engineer Walter Porstmann with the metric system adopted as DIN 476 in 1922: a one-square-meter A0 sheet, a constant 1:√2 ratio, and smaller formats made by halving. The principles became international through ISO 216 in 1975; the current ISO 216 edition specifies trimmed A- and B-series paper sizes.
A4's dimensions look awkward because the design is solving for behavior, not pretty numbers. It is a rectangle chosen for what happens after the scissors come out.