Golden Ratio Calculator
Split any length into golden-ratio proportions. Tell it which piece you know — the longer segment (a), the shorter segment (b), or the whole (a + b) — and it fills in the other two so that a/b = (a+b)/a = φ ≈ 1.618. Any unit works.
Example: with What do you know? Longer segment (a) · Value 10 → Shorter segment (b): 6.18 units.
- Longer segment (a)10 units
- Whole (a + b)16.18 units
- Golden checka/b = (a+b)/a = 1.618 = φ
Computed by the calculator below using its default values. Change any input to see your own numbers.
φ = (1 + √5)/2 = 1.6180339887… — the only positive number where a/b equals (a+b)/a. Consecutive Fibonacci numbers approximate it.
What makes 1.618 special
The golden ratio is the one way to cut a length so the whole relates to the large piece exactly as the large piece relates to the small one: (a+b)/a = a/b. Setting that common ratio to x gives x² = x + 1, whose positive root is φ = (1 + √5)/2 = 1.6180339887…. That self-similarity is the whole trick — a golden rectangle with a square sliced off leaves another golden rectangle, forever.
Fibonacci numbers sneak up on φ: 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 34/21 ≈ 1.619, converging as the numbers grow. Designers use the ratio for layout splits (a 1000 px canvas divides at about 618 px), photographers as a softer alternative to the rule of thirds, and woodworkers for frame proportions. Claims that φ governs seashells and human faces are mostly folklore — treat it as a pleasing convention, not a law of nature.
How it’s calculated
φ = (1 + √5)/2 ≈ 1.6180339887. Given the longer piece a: b = a/φ and whole = a·φ (because 1 + 1/φ = φ). Given the shorter piece b: a = b·φ, whole = b·φ². Given the whole W: a = W/φ, b = W − a = W/φ². Every mode keeps a/b = (a+b)/a = φ.
A design convention, not a natural law — use it because the proportions look right to you, not because it is mathematically superior to other ratios.
Fibonacci ratios closing in on φ
| Pair | Ratio | Off from φ by |
|---|---|---|
| 3 : 2 | 1.5 | −0.118 |
| 5 : 3 | 1.667 | +0.049 |
| 8 : 5 | 1.6 | −0.018 |
| 13 : 8 | 1.625 | +0.007 |
| 21 : 13 | 1.6154 | −0.0026 |
| 34 : 21 | 1.6190 | +0.0010 |
Ratios of consecutive Fibonacci numbers; they alternate above and below φ = 1.6180339887 and converge toward it.
Common mistakes
- Multiplying the whole by 1.618 to find the larger piece — the larger piece is the whole divided by φ (61.8% of it), not more than the whole.
- Using 1.6 or 1.62 in chained calculations; rounding error compounds, so keep φ = 1.618034 until the final rounding.
- Confusing φ (1.618) with 1/φ (0.618) — dividing by φ and multiplying by 0.618 are the same thing, so do one, not both.
- Expecting a/b and (a+b)/a to match at any other ratio; the equality is what defines φ and holds nowhere else.
Frequently asked questions
What is the golden ratio formula?
φ = (1 + √5)/2 ≈ 1.6180339887. It is the positive solution of x² = x + 1, which encodes the defining property that the whole-to-large ratio equals the large-to-small ratio.
How do I split a line in the golden ratio?
Divide the total length by φ to get the longer piece; the remainder is the shorter piece. A 100-unit line splits at 61.803 and 38.197 — the classic 61.8% / 38.2% proportions.
What is a golden rectangle?
One whose long side is φ times its short side, like 16.18 × 10. Remove the largest possible square and the leftover rectangle is golden again, which is how the golden spiral drawings are constructed.
Is 1.618 the same as 0.618?
They are reciprocals: 1/φ = φ − 1 = 0.6180339887. Both appear in design specs, so check whether a source means "multiply by 1.618" or "take 61.8% of" — they scale in opposite directions.
Do Fibonacci numbers really give the golden ratio?
Ratios of consecutive Fibonacci numbers (8/5, 13/8, 21/13…) approach φ ever more closely but never equal it exactly — φ is irrational. By 34/21 the gap is already about 0.001.