An A4 sheet measures 210 by 297 millimetres. Neither number looks especially tidy. Why not make it 200 by 300 instead?
Try imagining that simpler rectangle cut across its middle. Each half would measure 150 by 200. Turn one upright and it looks broader than the original. The area has halved, but the shape has changed.
A4 belongs to a family designed around a different result: the smaller rectangle should have the same proportions as the larger one. That is the rule behind the A-series. There is a small qualification, though. The mathematical shape is exact; the dimensions printed on a packet of paper use whole millimetres.
The cut that sets the shape
Picture a portrait sheet with a short side of one unit and a long side of r units. Cut across the long side at its midpoint. Each half has sides of one and r/2. Turn a half upright: its long side is now one, and its short side is r/2.
For the original and the half to share a shape, their long-to-short ratios must match. That gives r = 2/r, or r squared = 2. So r must be the square root of two: about 1.4142.
The University of Cambridge's NRICH paper-size investigation develops this relationship through similar rectangles. The square root is the consequence of asking the shape to survive the cut. It is not a decorative number chosen afterwards.
A square cannot do the same trick. Halve a square and the pieces are rectangles twice as long as they are wide. Halve those across their longer dimension and squares return. The square-root-of-two rectangle avoids that alternating shape.
Why 210 by 297?
A ratio determines a shape, but it cannot determine a size. A tiny rectangle and an enormous one can have identical proportions. The A-series needs a starting area as well.
That starting point is A0, with a theoretical area of one square metre. Four successive halvings give A1, A2, A3 and A4. In the ideal geometry, A4 therefore has one-sixteenth of a square metre: 0.0625 square metres.
Combine that area with the square-root-of-two ratio and the calculated A4 sides are approximately 210.224 and 297.302 millimetres. The familiar dimensions are 210 by 297. Markus Kuhn's explanation and dimension table makes the essential distinction: the system uses the ratio, while standard sheet dimensions are expressed in rounded millimetres.
This produces a revealing detail. Half of a nominal 297-millimetre side is 148.5 millimetres, but the listed A5 size is 148 by 210. Cutting a nominal A4 sheet exactly in half therefore does not produce two pieces measuring precisely the listed A5 dimensions. Saying that A4 halves into A5 is a useful description of the system, rather than an assertion that every physical measurement remains mathematically exact.
Half the area is not half the height
The same geometry explains a photocopier setting that can otherwise look odd. To reduce a page to half its area while preserving its shape, both side lengths must become about 70.71 per cent of their originals. Multiply 0.7071 by itself and the result is approximately 0.5.
Reducing both sides to 50 per cent would instead leave a quarter of the area. The distinction is between shrinking a length and shrinking a surface. NRICH's worked scaling explanation shows why the A3-to-A4 reduction is about 70.7 per cent.
In the ideal model, a page can shrink into the next size without needing a different shape. That does not make its small text easier to read, or make a printer's margins disappear. The geometry preserves proportions; it does not promise identical results from every machine or layout.
An old observation became a shared system
The idea predates the modern standard. In a letter to Johann Beckmann dated 25 October 1786, Georg Christoph Lichtenberg described a teaching problem about paper formats retaining their proportions. He also reported finding the desired proportions in paper already available to him. His letter is evidence of the observation, rather than proof that he invented every part of the later system.
DIN's history of the A-series credits engineer Walter Porstmann with developing the system around the one-square-metre base. DIN published DIN 476 on 18 August 1922. Its account also describes how the formats influenced envelopes, folders, filing trays and furniture: agreeing on paper affected the things built to hold it.
The modern ISO 216 standard covers trimmed A- and B-series paper sizes. A mathematical property became something people and equipment could share.
That is what the apparently awkward 210 and 297 are doing. They connect an everyday sheet to a larger family, with a useful relationship between each size and the next. The numbers are rounded. The idea behind them is remarkably precise.
Sources
Cambridge NRICH: geometry and scaling; Markus Kuhn: dimensions and rounding; Lichtenberg's letter, with translation; DIN: standardisation history; ISO 216: scope of the standard.