The ratio that survives a fold
A4 belongs to the ISO A paper series, whose sizes are built around one simple rule: each sheet has a long-to-short side ratio of √2, approximately 1.414. That unusual proportion is what lets the shape repeat across sizes.
Halve an A4 sheet across its longer dimension and each half is A5. The new rectangle’s sides are the original short side and half the original long side. Because the long side was √2 times the short one, halving it gives a side that is 1/√2 of the short side. The ratio of the half-sheet’s longer side to its shorter side is therefore √2 again.
Why designers and printers care
Many familiar rectangles change shape when resized: make one half as wide and it becomes relatively taller or flatter. With the A-series ratio, a page can be reduced or enlarged to the next standard size without changing the proportions of its contents. A diagram, photograph, or page layout can be scaled uniformly rather than squeezed in one direction.
That consistency also makes paper sizes practical for copying and printing. A document designed for one A size can be reduced to the next smaller size, or enlarged to the next larger one, while keeping its original shape. The image may need to be scaled, but it does not need to be redesigned merely to avoid distortion.
The rule’s memorable consequence is geometric: cut an A-series sheet in half, and each half has the same shape as the whole.
