The 1:√2 aspect ratio: the maths behind A4

Why √2 of all numbers?

Take a sheet with a short side of 1 and a long side of x. Cut it in half across the long side and the new sheet has sides of x/2 and 1. For both sheets to have the same shape:

x : 1 = 1 : x/2

It follows that x² = 2, so x = √2 ≈ 1.41421. There is no other solution. Any other rectangle changes its shape when halved, becoming either slimmer or squatter.

How to calculate any A size

Porstmann also specified that A0 has an area of one square metre. With a as the short side, a × a√2 = 1 m². From that:

A0

short side = 1 / ⁴√2 m ≈ 0.8409 m, long side = ⁴√2 m ≈ 1.1892 m

An in general

short side = 2^−(2n+1)/4 m, long side = 2^{−(2n−1)/4} m

Example A4 (n = 4)

2^−9/4 m = 0.2102 m and 2^−7/4 m = 0.2973 m, rounded to 210 × 297 mm

B series

B0 has a short side of 1 m, Bn = 2^−n/2 m × 2^−(n−1)/2 m

C series

the geometric mean of the A and B sizes with the same number

The standard rounds down to whole millimetres. That is why A5, at 148 mm, is slightly shorter than the exact half of A4 (148.5 mm).

The copy factors

Because all sizes share the same shape, any size can be scaled to any other without margins and without cropping. One step up means a factor of √2, one step down 1/√2.

The full table for every combination from A0 to A10 is in the scaling factor calculator.

Other aspect ratios compared

ISO sizes therefore sit between the squat Letter and the elongated golden ratio. You often read that A4 follows the golden ratio. It does not: 1.414 and 1.618 are clearly different.

A way to check it yourself

Fold an A4 sheet so that the short edge lies along the long edge. You get a triangle and a square. The diagonal of that square, which is the folded edge, is exactly as long as the long side of the sheet. That is Lichtenberg’s description from 1786, and you can try it with any sheet from your printer.

Frequently asked questions