Grades 6–8 · Mathematics
Why does A-series paper keep its shape when halved?
A folded sheet can become a booklet. In the A-series paper family, including A4 and A5, halving a sheet across its long dimension makes two smaller sheets with almost the same proportions as the original. Turning one half upright reveals the connection.
In the diagramHalve the long side, then turn the halfOriginal sheetOne half, turnedRatio 1.41Half the area√2 ≈ 1.414: the two ratios agree.
Call the original short side a and long side b. The long-to-short ratio is b/a. A half has sides a and b/2, so its ratio after turning is a/(b/2), or 2a/b. To keep the shape, these ratios must agree. Setting r = b/a gives r = 2/r, so r² = 2 and r = √2, about 1.414.
The half has half the area, but its lengths are not both halved. Corresponding lengths shrink by 1/√2, about 0.707. This lets a page design scale between sizes while keeping its proportions. Actual A-series dimensions use rounded millimeters, so the physical ratios are approximate. The picture uses an exact ideal rectangle to explain the geometry; other paper formats can have different proportions.
About this explanation
Starting knowledge: Understands ratios, area and square roots.
Sources
- University of Cambridge NRICH — Fit for Photocopying
- International Organization for Standardization — ISO 216:2007 — A and B series paper sizes