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by potash 4506 days ago
Interesting, I'd never thought about this before. In case you (or anyone else) are still wondering...

If an A3 sheet is X units by Y units (with X the long side) and A4 is 2Y units by X units then their ratios are X/Y and 2Y/X, respectively. If these are equal we have

2Y/X = X/Y

cross multiplying gives

2Y^2 = X^2

rearranging and taking a square root (X and Y are positive) gives

X/Y = sqrt(2).