Golden Ratio Calculator
Divide a length in the golden ratio, or find the whole from a part — φ = 1.6180339887.
Two lengths are in the golden ratio when the whole is to the longer part as the longer part is to the shorter: (a+b)/a = a/b.
How the golden ratio calculator works
Two lengths are in the golden ratio when the whole is to the longer part as the longer part is to the shorter: (a+b)/a = a/b. That condition has one positive solution, φ = (1+√5)/2 = 1.6180339887…
φ has a property no other number shares: φ² = φ + 1, and 1/φ = φ − 1. So its square and its reciprocal are both just φ shifted by one, which is why it turns up whenever something grows by adding its own previous size.
Enter a length and this splits it into golden sections, or enter one part and it gives the other and the whole.
Formula: φ = (1 + √5) / 2; longer = whole / φ; shorter = whole − longer
Worked examples
| Inputs | Longer part (a) | Note |
|---|---|---|
| Split 100 in the golden ratio | 61.803399 | 61.80 and 38.20 |
| A longer part of 61.8 — what is the whole? | 61.803399 | 100 |
| From the shorter part | 61.803399 | longer 61.80, whole 100 |
FAQFrequently asked questions
What is the golden ratio?
φ = 1.6180339887…, the number for which the whole is to the larger part as the larger part is to the smaller.
How do I split a length by the golden ratio?
Divide it by 1.618. A 100 mm line splits into 61.80 and 38.20.
Why is φ² = φ + 1?
It follows directly from the defining equation; φ is the positive root of x² − x − 1 = 0.
Is the golden ratio really in art and nature?
Genuinely in phyllotaxis and in some deliberate designs. Many famous claims — the Parthenon, the Mona Lisa — are retrofitted and do not survive measurement.
How does it relate to Fibonacci?
Consecutive Fibonacci numbers approach φ when divided, more closely with every term.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.