Golden Ratio Calculator

Solve golden ratio proportions instantly: enter the whole, long, or short length and get the other two via φ = 1.618. Shows φ, 1/φ, φ² and the 61.8/38.2 split.

Golden Proportion

Short segment (b)

100

Known length → other sides via φ = 1.61803398

Whole (a + b)261.803399
Long (a)161.803399
Short (b)100
Split61.8% / 38.2%

Known Length

whole = long + short; the proportion (a+b)/a = a/b = φ is enforced.

Any positive number — pixels, inches, cm; results use the same unit.

The Proportion

(a + b) / a = 261.803399 / 161.803399 = 1.61803399

a / b = 161.803399 / 100 = 1.61803399

both equal φ ≈ 1.6180339887 ✓

a = 161.803399
b = 100

Golden Ratio Reference Values

Quick Answer

The golden ratio φ = (1+√5)/2 ≈ 1.6180339887 splits a length so that whole/long = long/short = φ. Enter any one of the three lengths — whole (a+b), long (a), or short (b) — and the other two are computed instantly: whole = long + short, long = short × φ, so a 100-unit short gives a 161.8-unit long and a 261.8-unit whole. Designers use it for layout grids, logos, and typography scale.

Key Facts

  • φ = (1+√5)/2 = 1.6180339887… — an irrational number, also called the golden mean
  • Defining property: (a+b)/a = a/b = φ — the whole relates to the long part as the long part to the short
  • 1/φ = φ − 1 = 0.6180339887… — the only positive number that is its own reciprocal minus one
  • φ² = φ + 1 = 2.6180339887… — squaring φ is the same as adding 1 to it
  • Splits: long = short × φ; whole = short × φ² = long × φ
  • Fibonacci ratios converge to φ: successive terms 1, 1, 2, 3, 5, 8, 13, … approach 1.618 within ~10 steps
  • In degrees, the golden angle is 360°/φ² ≈ 137.5° — the arrangement of sunflower seeds and pinecones
  • Common uses: layout grids (61.8/38.2 split), logo construction, typographic scale factors, aspect ratios

Frequently Asked Questions

It is the unique positive number φ where a length divided into two parts satisfies: the whole divided by the long part equals the long part divided by the short part — both equal φ. Algebraically (a+b)/a = a/b, which solves to a/b = (1+√5)/2 ≈ 1.6180339887. It is irrational: φ cannot be written as a fraction of integers, and its decimal never repeats.

Depends on which length you know. Given the short side b: long = b × 1.618, whole = b × 2.618. Given the long side a: short = a ÷ 1.618, whole = a × 1.618. Given the whole: long = whole × 0.618, short = whole × 0.382. This calculator solves for the missing two lengths automatically once you pick which one you entered.

From the defining equation φ² = φ + 1, divide both sides by φ: φ = 1 + 1/φ, which rearranges to 1/φ = φ − 1 = 0.6180339887. This self-similar property is what makes the golden spiral work: removing a square from a golden rectangle leaves a smaller golden rectangle, and the ratio between successive remains is always 0.618.

The ratio of successive Fibonacci numbers converges to φ: 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 ≈ 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 55/34 ≈ 1.6176, 144/89 ≈ 1.6180. The convergence is exact in the limit — Binet’s formula expresses the n-th Fibonacci number as (φⁿ − (−φ)⁻ⁿ)/√5.

It genuinely appears in phyllotaxis (seed heads pack at the golden angle 137.5° because it is the most irrational — least oversamplable — rotation) and in continued-fraction theory. Many popular claims, though, are retrospective: measurements of the Parthenon, the Mona Lisa, or the human face can be tuned to hit 1.618 only by choosing convenient points. Treat design uses as a strong aesthetic convention rather than a law of beauty.

Layout: split a 1440px canvas at 61.8/38.2 (≈890/550px) for content/sidebar proportions. Typography: multiply a base font size by φ for a heading scale (16 → 25.9px). Logo geometry: construct circles whose radii step by 1/φ. Photography: place the subject on a 1.618 gridline instead of the rule-of-thirds 1.5 line for a slightly tighter composition.