Silver Ratio Calculator

Divide or scale a length by the silver ratio (1+sqrt2 ≈ 2.414) or the paper ratio 1:sqrt2, with a silver rectangle diagram.

Input

Divide or scale a length by the silver ratio. Choose the type, then enter the known length and whether it is the short or long side.

Known length

This length is

The long side to short side ratio is 1+sqrt2. The dashed line marks where a square equal to the short side is removed.

Silver ratio

2.41421356

Long side

241.42135624

Short side

100

Side of removed square

100

Long side of remaining rectangle

141.42135624

Area

24,142.13562373

Diagonal

261.31259298

In a metallic silver rectangle, removing two squares equal to the short side leaves a rectangle similar to the original.

The golden ratio is about 1.61803399, and the gap from this ratio is about 0.79617957.

The silver ratio has two common meanings. As a metallic ratio it is delta = 1 + sqrt2 ≈ 2.414, the value whose continued fraction is a repeating sequence of twos. In design and papermaking the silver ratio (paper ratio) instead means 1:sqrt2 ≈ 1.414. This tool lets you switch between both.
A silver rectangle with ratio delta = 1 + sqrt2 is self similar: removing two squares whose side equals the short side from the long side leaves a rectangle similar to the original. The dashed line in the diagram marks the boundary of the removed square.
A rectangle with the paper ratio 1:sqrt2 stays 1:sqrt2 when you fold its long side in half. A and B paper series (such as A4) use this ratio, so the aspect ratio is preserved under scaling.
The golden ratio phi ≈ 1.618 is often confused with the silver ratio but is different. The golden rectangle stays similar after removing one square, while the metallic silver ratio stays similar after removing two squares. This tool also shows the gap from the golden ratio.
Area is short side times long side, and the diagonal comes from the short and long sides via the Pythagorean theorem. Values are shown to eight decimal places.

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