ISO 216 · desk geometry
How one square metre becomes A4
The A series starts with A0 and its nominal area of 1 m². Four halvings produce A4, while the 1:√2 ratio makes the same shape reappear at every step.
Dimensions published by DIN under ISO 216:2007.
A0 → A1 → A2 → A3 → A4 210 × 297 mm
Try the foldThat is all the nominal square metre loses when A0 sides are rounded to 841 × 1,189 mm.
Folding press
Choose a size; the proportion stays
Each step halves the ideal area. Whole-millimetre dimensions are rounded, so the departure grows gradually in the smaller sizes.
- Size
- A4
- Published dimensions
- 210 × 297 mm
- Ideal area
- 0.0625 m²
- Area from dimensions
- 0.062370 m²
- Departure from ideal
- −130 mm² · −0.208%
- From one ideal A0
- 16 sheets of A4
The ratio
One proportion solves the fold
Call the long side L and the short side S. The starting ratio is r = L/S. After halving and rotation, the ratio becomes 2/r.
r = 2/r → r² = 2 → r = √2
This positive ratio is unique. A drawing or a page of text can move between sizes without changing proportions, subject to printable margins.
Millimetre traces
The ideal shape survives; whole numbers leave small departures
A0 has a nominal area of 1 m². Multiplying its published dimensions gives 0.999949 m², or 0.0051% less. At A4, the area departure reaches 0.208%; at A8 it approaches 1.5%.
| Size | Dimensions | Ideal area | Calculated area | Departure |
|---|---|---|---|---|
| A0 | 841 × 1,189 mm | 10,000.00 cm² | 9,999.49 cm² | −0.0051% |
| A1 | 594 × 841 mm | 5,000.00 cm² | 4,995.54 cm² | −0.0892% |
| A2 | 420 × 594 mm | 2,500.00 cm² | 2,494.80 cm² | −0.2080% |
| A3 | 297 × 420 mm | 1,250.00 cm² | 1,247.40 cm² | −0.2080% |
| A4 | 210 × 297 mm | 625.00 cm² | 623.70 cm² | −0.2080% |
| A5 | 148 × 210 mm | 312.50 cm² | 310.80 cm² | −0.5440% |
| A6 | 105 × 148 mm | 156.25 cm² | 155.40 cm² | −0.5440% |
| A7 | 74 × 105 mm | 78.125 cm² | 77.70 cm² | −0.5440% |
| A8 | 52 × 74 mm | 39.0625 cm² | 38.48 cm² | −1.4912% |
The copier scale
√2 becomes the 141.4% button
Moving up one size multiplies each side by √2, or 141.4%. Moving down uses the inverse ratio, 70.7%. These values preserve a drawing's proportions between A4 and A3.
Sheet mass
The geometry weighs about 2.495 kg in a ream of A4
Grammage states mass per square metre. Using the published A4 dimensions, 80 g/m² gives 4.9896 g per sheet and about 2.495 kg for 500 sheets before packaging.
The calculation uses the area from the published dimensions. Moisture, manufacturing tolerances, coatings, and packaging can change measured mass.
Scope of the standard
ISO 216 covers trimmed writing paper and certain printed matter
ISO 216:2007 applies to trimmed paper for administrative, commercial, and technical use, as well as certain forms and catalogues. Newspapers, published books, posters, and special items may use other standards.
DIN published the A-series rule as DIN 476 on 18 August 1922. The dimensions later entered the international standard.
How the values are calculated
Method and sources
A0–A8 dimensions come from the DIN table. Calculated area is width × height; ideal area is 1/2ⁿ m² for size An. The departure divides the difference by ideal area. Mass uses calculated area and the selected grammage; rounding occurs only for display.
- ISO 216:2007 for edition and scope.
- DIN, Happy birthday, A4! for dimensions, mechanism, and history.
- University of Cambridge, International standard paper sizes for an independent check of the geometry and scales.
Sources accessed 29 August 2026.