The zoom buttons on a copier show oddly specific numbers: 141%, 122%, 115%, 71%. They look arbitrary, but nobody chose them for convenience — they fall out of the paper standards themselves. Once you know how, you can work out ratios that are not on any button (fitting a B5 original onto A3, say). This article lists the A- and B-series dimensions and shows exactly how the ratios are derived.
The A series is defined by "one square metre" and "1:√2"
The A series starts at A0, whose area is exactly one square metre. On top of that the aspect ratio is fixed at 1:√2, so folding the long side in half produces the next size with the same proportions. Fold A0 (841 × 1189 mm) and you get A1 (594 × 841 mm); fold again for A2, and so on.
Japan's B series works the same way but starts from an area of 1.5 square metres. That is why sizes interleave in the sequence A4, B4, A3.
| Number | A series | A area | B series (JIS) | B area |
|---|---|---|---|---|
| 0 | 841 × 1189 mm | 1.000 m² | 1030 × 1456 mm | 1.500 m² |
| 1 | 594 × 841 mm | 0.500 m² | 728 × 1030 mm | 0.750 m² |
| 2 | 420 × 594 mm | 0.249 m² | 515 × 728 mm | 0.375 m² |
| 3 | 297 × 420 mm | 0.125 m² | 364 × 515 mm | 0.187 m² |
| 4 | 210 × 297 mm | 0.062 m² | 257 × 364 mm | 0.094 m² |
| 5 | 148 × 210 mm | 0.031 m² | 182 × 257 mm | 0.047 m² |
| 6 | 105 × 148 mm | 0.016 m² | 128 × 182 mm | 0.023 m² |
Every ratio is the square root of an area ratio
Enlargement is specified in side length, not area. To double the area you multiply the sides by √2; to multiply the area by 1.5 you multiply the sides by √1.5. So there are really only two numbers to remember.
- One step within a series (A4↔A3, B5↔B4): area ×2, so √2 = 1.4142…
- Between A and B (A4↔B4, A3↔B3): area ×1.5, so √1.5 = 1.2247…
- Two steps (A4↔A2): area ×4, so exactly 2.
Computed from the actual dimensions, the ratios come out as follows. The small difference between the short and long sides comes from the standard sizes being rounded to whole millimetres. "Fits" is the smaller of the two rounded down to a whole percent (this value always fits); "Nearest" is simply rounded to the closest whole percent.
| Conversion | Short side | Long side | Fits | Nearest |
|---|---|---|---|---|
| A4 → A3 | 141.43% | 141.41% | 141% | 141% |
| A3 → A4 | 70.71% | 70.71% | 70% | 71% |
| A4 → A2 | 200.00% | 200.00% | 200% | 200% |
| A4 → B4 | 122.38% | 122.56% | 122% | 122% |
| B4 → A4 | 81.71% | 81.59% | 81% | 82% |
| B5 → A4 | 115.38% | 115.56% | 115% | 115% |
| A4 → B5 | 86.67% | 86.53% | 86% | 87% |
| A3 → B5 | 61.28% | 61.19% | 61% | 61% |
Three things stand out.
1. 141% and 71% are opposites, but they do not add up to 200%
A4 to A3 is 141%; A3 to A4 is 71%. Added together that is 212%, which does not look like a there-and-back pair. That is because ratios live in the world of multiplication: 1.4142 × 0.7071 = 1.0000, and multiplying them returns you to the original. Enlarging and then reducing by "the same number" fails precisely when you reason about it additively.
2. "71%" is the rounded value, and it overshoots by 0.87 mm
The computed ratio for A3 to A4 is 70.71%. Rounded to the nearest percent that is 71% — but at 71%, A3's 297 mm short side becomes 210.87 mm, overshooting A4's 210 mm by 0.87 mm. The same happens across the series: B4 to A4 at 82% gives 257 × 0.82 = 210.74 mm (0.74 mm over), and A4 to B5 at 87% gives 210 × 0.87 = 182.70 mm (0.70 mm over). With margins you will never notice, but artwork or rules running to the edge get clipped. When it must fit, type in the rounded-down value (70%, 81%, 86%).
3. Short and long sides differ, so use the smaller one
A4 to B4 is 122.38% on the short side and 122.56% on the long side. Taking the larger value (122.56%) makes the short side 210 × 1.22559 = 257.37 mm, overshooting B4's 257 mm by about 0.37 mm. Taking the smaller one (122.38%) leaves the long side at 297 × 1.22381 = 363.47 mm, inside B4's 364 mm. Which side the slack lands on is not fixed, so compute both and take the smaller.
Aside: not every size is exactly 1:√2
Rounding to whole millimetres shifts the ratio slightly from size to size. Computed out, A4 is 1.41429, A5 is 1.41892 and A6 is 1.40952, against √2 at 1.41421 — so A5 is about 0.33% taller and A6 about 0.33% wider in proportion (A4 is off by 0.005%, which you can ignore). "All A sizes are the same shape" is not strictly true. Where exact dimensions matter, such as borderless printing or die cutting, work from the millimetre figures rather than the ratio.
Where this trips people up
- You need a ratio that has no button: divide the target short side by the original short side. B5 onto A3 is 297 ÷ 182 = 163%.
- Combining two sheets into one: two A4 pages side by side make A3 exactly. Leave the zoom at 100%, load A3 and use 2-in-1 — no reduction is needed.
- Enlarging part of a document: the ratio applies to sides, not area, so 141% doubles the area. To make text twice as large you need 200%, which means paper two sizes up.
- Working across borders: the B series is not universal (see below). A "B5" in one program may not be the B5 you expect.
Careful: Japan's B series is not ISO B
Every B figure above is the JIS (Japanese Industrial Standards) B series, which starts at B0 = 1030 × 1456 mm = 1.5 m². ISO also defines a B series, but it starts at B0 = 1000 × 1414 mm, sized as a geometric midpoint between A sizes. In B5 terms, Japan is 182 × 257 mm and ISO is 176 × 250 mm — a difference of 6 to 7 mm. Pick B5 in software or a printer driver built around ISO and you get a page slightly smaller than a Japanese B5. The A series is shared by ISO and JIS, so sticking to A sizes is the safer choice for documents that cross borders.
FAQ
What ratio enlarges A4 to A3?
How do I work out ratios between the A and B series?
Why do the short and long sides give slightly different ratios?
Is B5 the same size everywhere?
Dimensions are the nominal sizes of the A series (ISO 216 / JIS P 0138) and Japan's B series (JIS P 0138). Areas, aspect ratios and zoom ratios are computed from those nominal sizes (areas to three decimal places, ratios to two). The values printed on a copier's zoom buttons are rounded differently by different machines. As of 2026-08-10.