Golden Calculator
Golden calculator works out Golden ratio (phi) and 1 / phi from Its measurement, Larger number (a) and Smaller number (b). The default figures on this page give longer segment (a) of 61.803399; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
4 steps
- phi = (1 + sqrt 5) / 2 = 1.61803398875, and phi^2 = phi + 1 = 2.61803398875.
- Longer part = total / phi = 100 / 1.61803398875 = 61.803399.
- Whole = 61.803399 + 38.196601 = 100.
- Check: (a + b) / a = 1.6180339887 and a / b = 1.6180339887 - the defining equality of Euclid VI.3.
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
What the golden calculator calculates
Cuts a length in the golden ratio - give any one of the whole, the longer part or the shorter part and it returns the other two exactly - and measures how far any pair of numbers sits from phi.
How to use this golden calculator
Enter Its measurement, Larger number (a) and Smaller number (b). Golden ratio (phi) updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Golden ratio (phi), 1 / phi and Longer segment (a).
The method follows Euclid, Elements, Book VI definition 3 and proposition 30 - a straight line cut in extreme and mean ratio and NIST Digital Library of Mathematical Functions / OEIS A001622 - the decimal expansion of phi = (1 + sqrt 5)/2 = 1.61803399....
Cutting a length of 100 in the golden ratio example
With What do you want to do?: segments, Which length do you know?: total, Its measurement: 100, the calculator returns Longer segment (a): 61.8033989, Shorter segment (b): 38.1966011, Ratio a : b: 1.61803399. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Golden ratio (phi), 1 / phi and Longer segment (a). The inputs you can vary are Its measurement, Larger number (a) and Smaller number (b).
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
What is the golden calculator?
Cuts a length in the golden ratio - give any one of the whole, the longer part or the shorter part and it returns the other two exactly - and measures how far any pair of numbers sits from phi. On this page it is computed from Its measurement, Larger number (a) and Smaller number (b) and shown as Golden ratio (phi), 1 / phi and Longer segment (a).
Why is 1.618 called the golden ratio?
Cuts a length in the golden ratio - give any one of the whole, the longer part or the shorter part and it returns the other two exactly - and measures how far any pair of numbers sits from phi. On the figures this page opens with — its measurement 100 — the longer segment (a) is 61.803399. Put your own figures in their place and the working updates beside the result.
How to calculate golden number?
Cuts a length in the golden ratio - give any one of the whole, the longer part or the shorter part and it returns the other two exactly - and measures how far any pair of numbers sits from phi. Open “How this is worked out” beside the result to see each step for your own figures.
What information do I need for the golden calculator?
You need Its measurement, Larger number (a) and Smaller number (b). The page opens with default figures that give longer segment (a) of 61.803399, so you can see the scale before entering your own.
What does this golden calculator show?
It shows Golden ratio (phi), 1 / phi, Longer segment (a), Shorter segment (b), Whole length (a + b), Ratio a : b, Difference from phi and Reading, with the working and any warnings for your entries.
Can I check how the golden calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.