Half the last

Each sheet is half the last

An A4 sheet is 210 by 297 millimetres. Two rules produce those numbers: the sides stand in the ratio 1 to √2, and the base sheet, A0, has an area of one square metre.

The one shape that survives a cut

Cut a sheet across its long side and you get two smaller sheets. For those halves to have the same proportions as the sheet you started with, the long side has to be the short side.

Why √2If the sides are 1 and r, each half is 1 by r/2 when turned on its side. Same shape means r/1 = 1/(r/2), so r² = 2 and r = 1.414…

Fix the area of the base sheet at one square metre and the sizes follow: A0 is 841 × 1189 mm. Each smaller size is the one before it halved and rounded down to a whole millimetre, which is why A4 is 210 × 297 and not something tidier.1

The practical payoff is that layouts scale without changing shape. Two A4 pages side by side make an A3, and reducing that to A4 takes a factor of 1/√2, about 70.7%, the familiar 71% on a photocopier. Going up two sizes doubles both sides; because of the rounding, nominal A2 doubled is 840 × 1188, a millimetre short of A0.

Cut it four times

Each sheet below is the preceding one cut in half across its long side, turned upright and trimmed to the next nominal size, all drawn at one scale. The dashed line marks the next cut.

ISO A

  1. A4210 × 297 mm
    1 : 1.414
  2. A5148 × 210 mm
    1 : 1.419
  3. A6105 × 148 mm
    1 : 1.410
  4. A774 × 105 mm
    1 : 1.419
  5. A852 × 74 mm
    1 : 1.423

US Letter

  1. 8½ × 11 in≈ 216 × 279 mm
    1 : 1.294
  2. 5½ × 8½ in≈ 140 × 216 mm
    1 : 1.545
  3. 4¼ × 5½ in≈ 108 × 140 mm
    1 : 1.294
  4. 2¾ × 4¼ in≈ 70 × 108 mm
    1 : 1.545
  5. 2⅛ × 2¾ in≈ 54 × 70 mm
    1 : 1.294

A sizes keep the ratio; the small drift in the third decimal is the rounding to whole millimetres. Letter alternates between two shapes, 1 : 1.294 and 1 : 1.545, so a layout made for one size can’t fill its half without extra margins or cropping.

B sits outside A. C sits between.

Each B size is the of two neighbouring A sizes, which puts B4 halfway, in proportion, between A4 and A3. Each C size is the geometric mean of the A and B sizes with the same number.1

Geometric meanThe square root of the product of two numbers. For sides: B0’s short side is √(841 × 1189) ≈ 1000 mm.

Because C sits between A and B, each C envelope is just larger than the A sheet with the same number: a C4 takes an A4 sheet flat, a C5 takes it folded once, and a C6 folded twice.1

Choose a sheet to see its series.

An idea older than the standard

  1. Georg Christoph Lichtenberg writes to Johann Beckmann that a sheet whose sides stand as 1 to √2 can be halved and keep its shape. The letter is one of the oldest written records of the √2 ratio for paper, and Lichtenberg notes that paper of that shape was already commonly available.2
  2. A French tax law on stamped paper, the Loi sur le timbre, defines formats that match today’s ISO sizes.1
  3. Walter Porstmann’s proposal is adopted in Germany as DIN 476.1
  4. The system becomes the international standard ISO 216, and the United Nations document format.1

The holdouts are few. Kuhn’s reference lists the United States, Canada and, in part, Mexico as the industrialised countries where ISO sizes are not yet in wide use.1