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📏 Productivity & Units

A4 to A3 Is 141.42% Because the Whole System Is Built on a Square Root

Every A size has the same shape, so halving one gives the next. Letter does not share that shape, which is why no single scale fits both.

Scale to fit

 

Scale to fill

edges get cropped

Area change

Wasted on the sheet

after fitting

Both sizes

Reference

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How the calculation works

A4 and Letter, overlaid to scale A4 210 × 297 Letter 215.9 × 279.4 A4 is 5.9 mm narrower A4 is 17.6 mm taller so neither contains the other, and no single scale fills both. A4 → Letter fits at 94.07% Letter → A4 fits at 97.27%

How to Use This Tool

Pick the size a document was designed for and the size you are printing on. The scale-to-fit percentage is the number to type into the printer dialog.

Why the A series works the way it does

One rule generates the whole system: each size folded in half gives the next size down, with the same proportions. Only one ratio does that, and it is the square root of two — about 1.41421. A0 is then fixed by defining its area as one square metre.

The consequences are all useful:

  • Enlarging one step is always 141.42% and reducing one step is always 70.71%. Those are the two buttons on every office copier.
  • Two A4 sheets side by side make an A3 exactly, so booklet imposition needs no adjustment.
  • Area halves each step, which is why A4 is 1/16 of a square metre and why paper weight in grams per square metre converts so easily — an 80 gsm A4 sheet weighs exactly 5 grams.

Published dimensions are rounded down to whole millimetres at each step, which is why A4 is 210 × 297 rather than the exact 210.224 × 297.302. The ratio is therefore very slightly off at small sizes, and never enough to notice.

A4 and Letter cannot both fit

Letter is 215.9 × 279.4 mm, which is 8.5 × 11 inches. Its ratio is 1.294, not 1.414, so it is a fundamentally different shape.

A4 is 5.9 mm narrower and 17.6 mm taller. Neither sheet contains the other, so:

  • A4 onto Letter fits at 94.07%, limited by height.
  • Letter onto A4 fits at 97.27%, limited by width.

Every "fit to page" tick box silently applies one of those. Usually nobody notices, because the shrink is small and the margins absorb it — until something depends on absolute size, like a form designed to be measured, a scale drawing, or a document with content close to the edge.

Fold in half, get the next size A4 A4 A3 — 297 × 420 Same shape at every size, because the ratio is √2. A4 → A3   141.42% A3 → A4    70.71% The copier buttons are square roots, not round numbers.
No other ratio has this property, which is why the system was standardised on it.

Fit against fill

Two different scales, and printer dialogs are not always clear which one they use:

  • Fit uses the smaller of the two ratios, so everything is visible and one dimension has spare paper. This is what you almost always want.
  • Fill uses the larger, so the sheet is covered and the excess is cropped. Useful for photographs and destructive for documents.

Neither accounts for the printer's unprintable margin. Most printers cannot mark the outer few millimetres, so a document scaled to exactly 100% of the paper will still lose its edges. Genuine edge-to-edge printing requires a borderless-capable printer, or printing oversized and trimming.

The other systems

  • B series — geometric means between A sizes, so B4 sits between A3 and A4. Used for posters, passports and some books.
  • C series — envelopes. A C4 envelope holds an A4 sheet flat, and C5 holds A4 folded once. That is the entire logic.
  • US sizes — Letter, Legal, Tabloid and Ledger, defined in inches with no consistent ratio between them. Legal is Letter with 76.2 mm added to the height only, so it is a different shape again.
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Frequently Asked Questions

What percentage is A4 to A3?
141.42%, and A3 to A4 is 70.71%. Both are the square root of two, because every A size has the same proportions and each step doubles or halves the area.
Why is A4 210 by 297 mm?
Because the A series requires a side ratio of √2 so that folding a sheet in half gives the same shape, and A0 is defined as one square metre. Working down from there gives A4, with dimensions rounded down to whole millimetres at each step.
What scale prints A4 onto Letter?
94.07%, limited by height — A4 is 17.6 mm taller than Letter. Going the other way, Letter onto A4 is 97.27%, limited by width, since A4 is 5.9 mm narrower.
Why don't A4 and Letter fit each other?
They are different shapes. A4 has a ratio of 1.414 and Letter is 1.294, and A4 is both narrower and taller. Neither sheet contains the other, so there is no scale that fills both — every 'fit to page' setting shrinks by a few per cent.
What is the difference between fit and fill when printing?
Fit uses the smaller of the two scaling ratios so everything is visible, leaving spare paper on one dimension. Fill uses the larger so the sheet is covered and the excess is cropped. Fit is almost always what a document needs.
Why can't I print right to the edge?
Most printers have an unprintable margin of a few millimetres where the paper is gripped, so content scaled to exactly the sheet size still loses its edges. Edge-to-edge printing needs a borderless-capable printer, or printing oversized and trimming.

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