Golden Ratio Calculator
Apply the golden ratio φ ≈ 1.618 to any number: find its larger and smaller golden partner, or split a length into two golden segments — with the working shown.
- φ (phi)
- 1.6180339887
Calculator
About this calculator
The golden ratio, written φ (phi) and equal to (1 + √5) ÷ 2 ≈ 1.6180339887, is the proportion two quantities have when the larger is to the smaller as the whole is to the larger. This calculator applies it three ways: give it any number to find that number’s larger golden partner (value × φ) and smaller golden partner (value ÷ φ); give it a total length to split into a long and a short golden segment; or give it the long segment and recover the whole. Every result is unitless — work in pixels, centimetres, inches or pure numbers and the answers come back in the same unit.
How to read your results
The large figure at the top is the headline answer for the mode you chose — the larger partner, the long segment, or the reconstructed whole. The stats beneath restate every related quantity, and the subline confirms the golden relationship, for example that the larger partner divided by your value equals φ. In “Split a length” mode the long and short segments always add back to the total you typed, and their ratio long ÷ short is φ; in “From a segment” mode the whole divided by your segment is φ. The φ figure shown is the exact constant used, 1.6180339887, so you can check any of the divisions by hand.
How it's calculated
φ is the positive root of x² = x + 1, so φ = (1 + √5) ÷ 2 ≈ 1.6180339887 and its reciprocal 1 ÷ φ = φ − 1 ≈ 0.6180339887. Golden-partners mode multiplies and divides your value by φ: larger = value × φ, smaller = value ÷ φ. Split-a-length mode sets the long segment to total ÷ φ (equivalently 0.6180339887 × total) and the short segment to total − long, which guarantees long ÷ short = φ and total ÷ long = φ. From-a-segment mode treats your input as the long part: whole = long × φ and short = whole − long. The constant and these relationships follow Wolfram MathWorld and Wikipedia.
Worked example
Split a total length of 100 (Split a length mode).
The long segment is 100 ÷ φ ≈ 61.803, and the short segment is the remainder, 100 − 61.803 ≈ 38.197. Their ratio long ÷ short ≈ 61.803 ÷ 38.197 ≈ 1.618 = φ, and the total ÷ long ≈ 100 ÷ 61.803 ≈ 1.618 = φ — confirming a true golden cut.
Frequently asked questions
What is the golden ratio and what is φ exactly?
The golden ratio is the proportion in which the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part. It is the irrational number φ = (1 + √5) ÷ 2 ≈ 1.6180339887. A defining property is φ² = φ + 1, which also means 1 ÷ φ = φ − 1 ≈ 0.6180339887.
How do I split a length into the golden ratio?
Divide the total length by φ to get the long segment (the same as multiplying the total by about 0.618), then subtract it from the total to get the short segment. For a 100-unit line that gives a long part of about 61.803 and a short part of about 38.197, whose ratio is exactly φ.
What is the difference between the larger and smaller golden partner?
For a value A, the larger golden partner is A × φ (about 1.618 × A) and the smaller golden partner is A ÷ φ (about 0.618 × A). In each pair the bigger number divided by the smaller equals φ, so your value sits one golden step above the smaller partner and one step below the larger.
Does the calculator use specific units?
No — the golden ratio is a pure proportion, so the calculator is unitless. Whatever unit you have in mind for the input (pixels for a layout, millimetres for a print, or just a plain number) the outputs are in that same unit. Only the proportion matters.
Sources
Reviewed by the YouCalc Team · Last reviewed
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