Why A4 Paper Uses √2: ISO 216, Lichtenberg, B Sizes, and Letter

In October 1786, the German scientist Georg Christoph Lichtenberg wrote to Johann Beckmann about a sheet whose shape remained similar when halved. He calculated the ratio and observed that the writing paper in front of him already had approximately that proportion.

The letter is important evidence for the geometry and for an existing paper format, but it was not itself the modern ISO standard. Lichtenberg asked about the origin of the shape; later systems, legislation, industrial standards, and international standardisation were separate historical steps.

Let the long side be a and the short side be b. After halving the long side, the new rectangle has sides b and a/2. Similarity requires:

a / b = b / (a / 2)
a² = 2b²
a / b = √2 ≈ 1.41421356

For a positive non-square rectangle, √2:1 is the unique aspect ratio that remains similar under this halving operation. The theoretical area halves and the linear scale between adjacent sizes is 1/√2.

A French law of 1798 specified paper formats with dimensions that anticipate the metric, √2-based idea. Germany later standardised a systematic series as DIN 476 in 1922, associated with Walter Porstmann's work. The international A and B series were standardised as ISO 216 in 1975; the current ISO catalogue entry is ISO 216:2007.

These events should not be collapsed into a single invention. The mathematical observation, early legal dimensions, industrial series, and international standard each answer a different historical question.

The theoretical A0 rectangle has an area of 1 m² and an aspect ratio of √2:1. Successive sizes halve the area. Published ISO dimensions are rounded to whole millimetres, so their measured aspect ratios and areas are not mathematically exact.

This is why folding a physical 210 × 297 mm sheet gives 148.5 mm on one side while the nominal A5 width is 148 mm. The series preserves the theoretical relationship subject to the standard's rounding rules.

• A series: the principal ISO 216 series, beginning from theoretical A0 at 1 m².
• B series: in the unrounded construction, each size lies at the geometric mean of adjacent A sizes. Whole-millimetre nominal dimensions differ slightly from an exact geometric mean.
• C envelope series: in the unrounded construction, a C size lies between the corresponding A and B sizes so it can enclose the A sheet. ISO 269 specified C envelopes, but that standard was withdrawn in 2009. C4 and C5 remain in use through other standards and industry practice.

It is therefore inaccurate to describe A, B, and C as three current series all defined by ISO 216. ISO 216 covers the A and B trimmed-size series.

The United States and Canada commonly use Letter paper. ASME Y14.1 defines engineering drawing sheet sizes and formats rather than replacing general office-paper conventions.

Japan's JIS P 0138 was established in 1951. Its current 1998 edition is identified as modified in relation to ISO 216:1975, and the JIS B series differs from the ISO B series. “B5” therefore needs a standard or region when exact dimensions matter.

For the unrounded A series, the theoretical adjacent-size scale factors are √2 ≈ 141.4% and 1/√2 ≈ 70.7%. Two size steps correspond to 200% or 50% in aspect-ratio-preserving linear scaling.

Whole-millimetre nominal sizes do not have exactly the same horizontal and vertical ratio, so one percentage cannot match both edges perfectly. Printer non-printable areas, margins, bleed, binding allowance, and application rounding also affect the result.

Placing two A5 pages on A4 is geometrically convenient, but the √2 ratio is not itself a guarantee that a PDF booklet feature needs no margin or imposition calculation.

• The theoretical A-series rectangle uses the aspect ratio √2:1.
• Whole-millimetre standard dimensions approximate, rather than exactly reproduce, the theoretical ratios and areas.
• Lichtenberg's 1786 letter recorded the geometry and an existing sheet; DIN 476 and ISO 216 were later standardisation steps.
• ISO 216 covers A and B sizes, while the envelope standard ISO 269 was withdrawn in 2009.
• ISO B, JIS B, A4, and US Letter must be distinguished in practical workflows.