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Math

Calculate the Golden Ratio

Updated · Standard formula, shown below

Shorter part b 6.180 cm
Golden ratio φ
1.618034
Result
Whole length a + b
16.180 cm
Result
Longer part a
10 cm
You entered

This calculator splits a length in the golden ratio φ (about 1.618). Enter the longer part and you get the shorter part and the whole length, so that the whole is to the longer part as the longer part is to the shorter part.

Formula

Two lengths a > b are in the golden ratio when (a + b) ÷ a = a ÷ b = φ. Solving φ² = φ + 1 gives φ = (1 + √5) ÷ 2 ≈ 1.618034. So b = a ÷ φ and the whole length is a + b = a × φ.

  • a: the longer part (any unit)
  • b: the shorter part (same unit)
  • φ: the golden ratio, a pure number with no unit

Worked Example

Longer part a = 10 cm.

  1. φ = (1 + √5) ÷ 2 = (1 + 2.236068) ÷ 2 = 1.618034
  2. b = 10 ÷ 1.618034 = 6.180 cm
  3. Whole length = 10 + 6.180 = 16.180 cm (the same as 10 × 1.618034)

Limits and common mistakes

  • If you know the whole length L instead, the longer part is L ÷ φ and the shorter part is L ÷ φ².
  • φ is irrational; any decimal is rounded. Six decimals (1.618034) is enough for drawing, layout or woodwork.
  • The ratio is a proportion, so inches, centimetres or pixels all work as long as you keep one unit.
Formula used
in a | Longer part a | cm | 10 | 0
out b | Shorter part b | cm | a / ((1 + sqrt(5)) / 2) | 3
out phi | Golden ratio φ | - | (1 + sqrt(5)) / 2 | 6
out tot | Whole length a + b | cm | a + b | 3

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.