A4, Letter and the √2 Trick: How Paper Sizes Actually Work

You photocopy a document “two pages to a sheet,” and on A4 paper the result is flawless — two shrunken pages fill the sheet edge to edge with no awkward margin. Try the same trick on US Letter and you get a thin strip of white down one side. That difference isn’t a printer quirk; it’s baked into the geometry of the paper itself. The ISO A-series was engineered around a single mathematical property that US Letter simply doesn’t have, and once you see it, every “fit to page” surprise makes sense.

The √2 aspect ratio that makes A-series magic

Every sheet in the ISO 216 A-series has the same aspect ratio: the long side divided by the short side equals the square root of two, about 1.41421. This is the only ratio with a remarkable property — cut the sheet in half across its long dimension and each half keeps the exact same ratio. Fold A4 in half and you get two sheets shaped exactly like A5; A5 halves into A6, and so on, forever. Nothing distorts, nothing crops.

That’s why scaling between A sizes is lossless. Going from A4 to A3 doubles the area but preserves the proportions, so a document enlarges to fill the new sheet perfectly. The same √2 ratio is why two A5 pages tile an A4 sheet with zero waste. It’s a property the designers chose deliberately, and it cascades through the whole series.

Where A0’s odd dimensions come from

The numbers look arbitrary — A0 is 841 × 1189 mm — but they’re forced by two rules working together. First, A0 has an area of exactly one square metre. Second, it must have the √2 aspect ratio. Solve those two constraints simultaneously and you land on 841 × 1189 mm (rounded to whole millimetres). Every smaller size is then defined by halving the longer dimension:

A0  841 × 1189 mm   (1 m²)
A1  594 × 841 mm
A2  420 × 594 mm
A3  297 × 420 mm
A4  210 × 297 mm
A5  148 × 210 mm

Because A0 is one square metre, the area of any A size is easy to reason about: A1 is half a square metre, A2 a quarter, A4 is one sixteenth of a square metre. That single fact is also why paper weight is quoted in grams per square metre (gsm) — a sheet of 80 gsm A0 paper weighs exactly 80 grams, and an A4 sheet of the same stock weighs one sixteenth of that, about 5 grams.

The B and C series, and what envelopes are for

The A-series jumps in big steps — each size is 1.41× larger than the last in linear dimension — so ISO defines two companion series to fill the gaps. The B-series sits at the geometric mean of consecutive A sizes, giving you intermediate formats like B5 for books and passports. The C-series is sized to hold the A-series: a C4 envelope is built to swallow an A4 sheet flat, and a C5 envelope fits A4 folded once. That’s the system behind “C5 takes A5, or A4 folded in half” — the C-series exists almost entirely to make envelopes that match your documents.

Why US Letter breaks the spell

North American sizes come from a different tradition entirely, defined in inches rather than around a clean ratio:

US Letter   8.5 × 11 in    (215.9 × 279.4 mm)
US Legal    8.5 × 14 in    (215.9 × 355.6 mm)
US Tabloid  11 × 17 in     (279.4 × 431.8 mm)

US Letter’s aspect ratio is 11 ÷ 8.5 ≈ 1.294, not 1.414. So it doesn’t halve cleanly, and it doesn’t scale to A4 without either leaving a margin or cropping content. Tabloid is exactly two Letter sheets, which preserves that relationship, but the moment you cross between the inch world and the millimetre world the proportions stop lining up. This is the root cause of the classic “A4 document prints with cut-off edges on a US printer” headache.

Scaling one size onto another

When you need to print an A4 layout onto US Letter — or any size onto any other — the safe scale factor is the largest factor that fits the source inside the target without cropping. Compare the width ratio and the height ratio between the two sheets and take the smaller of the two:

fit = min(targetWidth / sourceWidth, targetHeight / sourceHeight)

For A4 (210 × 297 mm) onto US Letter (215.9 × 279.4 mm), the width ratio is about 1.03 but the height ratio is only about 0.94 — so you scale to 94% to keep everything on the page. That’s exactly the “shrink to fit” number print dialogs apply behind the scenes, and knowing it lets you predict the margins before you hit print.

Converting in practice

To convert any dimension between millimetres and inches, the only constant you need is 1 inch = 25.4 mm exactly. From there you can move any size into either system. Rather than memorise dozens of figures, the Paper Size Converter lists ISO A0–A10, the B and C series, Japanese JIS B sizes and the US formats side by side in mm, cm, inches and area — and it computes the fit-to-page scale factor between any two sizes you pick. If your work runs from screen to paper, pair it with the Pixels to cm / Inches Print Calculator to size an image correctly at a given DPI, and the Font Size Converter to keep type legible once it lands on the page. Learn the √2 trick once and the entire paper world stops being a guessing game.