Why Does A4 Paper Keep the Same Proportions When Folded in Half?

Fold an A4 sheet across the midpoint of its long side and it becomes A5. The smaller rectangle still looks like the original shape. This is deliberate: the side-length ratio is chosen to be the square root of two.

Let the short side be 1 and the long side be r. After halving, the new sides are 1 and r/2. Rotate the smaller sheet and its long side is 1 while its short side is r/2. To keep the original ratio, r must equal 1 divided by r/2, or 2/r. Therefore r squared equals 2, and the only positive solution is the square root of two. It is the unique ratio for which halving the area preserves the shape.

The A series in ISO 216 uses this property. A4 has nominal dimensions of 210 by 297 millimetres, and 297/210 is about 1.414, close to the square root of two; rounding to whole millimetres prevents infinite precision. Because the page proportions are approximately preserved, an A4 layout can be scaled to A5 as a whole without having to crop away one edge.

Here, “the same” means the same aspect ratio, not the same area, size or orientation. One irrational number turns a continuous geometric proportion into a paper system that can be halved repeatedly.

https://www.iso.org/standard/36631.html