What is the silver ratio?
The silver ratio is the proportion a : b = 1 : √2, where \(\sqrt{2} \approx 1.4142135624\). It is the construction ratio behind the international A-series paper sizes (A3, A4, A5) and appears throughout traditional Japanese design and architecture — it is sometimes called the "Yamato ratio" (Yamato-hi) or "hakugin-hi." A rectangle built on this ratio has a special property: cut it in half across its long side and each half is a smaller rectangle with exactly the same proportions. Although the idea is often presented through Japanese aesthetics, the math is universal: it is simply the constant \(\sqrt{2}\).
How to use this calculator
Pick which side you already know — the shorter side a (the "1") or the longer side b (the "√2"). Enter its length in any unit; the answer comes back in that same unit because a ratio is scale-free. Choose how many significant digits you want for the displayed result (the default is 10). The calculator then returns the matching side plus the full a : b pair.
The formula explained
If you know the short side a, the long side is $$b = a \cdot \sqrt{2}$$. If you know the long side b, the short side is $$a = \frac{b}{\sqrt{2}} = b \times 0.7071067812$$. Note this is the geometric silver ratio 1 : √2, not the algebraic "silver mean" \(\delta = 1 + \sqrt{2} \approx 2.4142\), which is a different constant.
Worked example: an A4 sheet
An A4 sheet has a short side of 210 mm. Multiply by √2: $$210 \times 1.4142135624 = 296.98 \text{ mm} \approx 297 \text{ mm}.$$ That matches the real ISO A4 dimension of 210 × 297 mm, confirming the formula.
FAQ
Does the unit matter? No. Whatever unit you enter (mm, cm, inches), the output is in the same unit, because the ratio has no dimension.
Is this the golden ratio? No. The golden ratio is \(\varphi \approx 1.618\). The silver ratio here is \(\sqrt{2} \approx 1.4142\).
Why offer so many significant digits? \(\sqrt{2}\) is irrational, so the answer never terminates. The digit selector lets you display as much precision as you need for engineering or design work.
