Paper sizes

The ISO 216 A series is built on one elegant constraint: every sheet has the same aspect ratio, and halving it across the long edge produces the next size down with that ratio preserved.

ratio = √2 ≈ 1.4142 A0 = 841 × 1189 mm = 1.000 m² A(n+1) = A(n) halved across the long side

Why √2 is the only possible answer

For a sheet of width w and height h to keep its proportions when halved, h/w must equal w/(h/2). Solving gives h/w = √2. No other ratio has this property, which is why the standard is not arbitrary — it is the unique solution.

What that buys you

Scaling between sizes never distorts anything: printing A4 content onto A3 is exactly 141%, and A4 onto A5 is 71%, with no cropping or letterboxing. Those two numbers are on every photocopier in the world outside North America.

Paper weights come free too

A0 is exactly one square metre, so an A0 sheet of 80 gsm paper weighs 80 grams — and an A4 sheet, being one sixteenth of it, weighs exactly 5 grams. That is why postal weight calculations are trivial in metric countries: a letter is simply five grams per sheet.

US Letter has no such logic

8.5 × 11 inches gives a ratio of 1.294, which does not survive halving. Scaling between Letter and Legal or Tabloid always crops or leaves margins. The origin is disputed — the most-repeated explanation involves the span of a vatman's arms in early paper-making, though the historical evidence for it is thin.

C series envelopes

The C series is the geometric mean of A and B, sized so that an A-series sheet fits inside the same-numbered C envelope. C4 takes an unfolded A4; C5 takes it folded once; C6 folded twice. DL is the odd one out — it predates the standard and takes an A4 folded into thirds.