Golden Ratio Calculator
Split a length by the golden ratio φ ≈ 1.618 — find the matching longer, shorter and total segments from any one of them.
How it works
φ = (1 + √5) ÷ 2 ≈ 1.6180339887 · total ÷ longer = longer ÷ shorter = φ
Two lengths are in the golden ratio when the whole is to the larger part as the larger part is to the smaller — a proportion equal to φ = (1 + √5) ÷ 2 ≈ 1.618. It has a unique self-referential property: φ = 1 + 1/φ, which is why dividing the total by φ gives the longer segment, and dividing the longer segment by φ gives the shorter. The ratio appears in art, architecture, design and the limit of the Fibonacci sequence.
Worked example
Take a longer segment of 100. The shorter segment is 100 ÷ φ ≈ 61.80, and the total is 100 + 61.80 ≈ 161.80 — which is also 100 × φ. Check the proportion: 161.80 ÷ 100 ≈ 1.618 and 100 ÷ 61.80 ≈ 1.618, the same golden ratio.
Sources: Golden ratio — Wikipedia
Frequently asked questions
What is the golden ratio exactly?
It’s the irrational number φ = (1 + √5) ÷ 2 ≈ 1.6180339887. It’s the only positive number where subtracting 1 gives its own reciprocal: φ − 1 = 1/φ.
How does φ relate to Fibonacci numbers?
The ratio of consecutive Fibonacci numbers converges to φ. 55 ÷ 34 ≈ 1.6176 and 89 ÷ 55 ≈ 1.6182 — closing in on 1.618 from both sides.
Where is the golden ratio actually used?
Designers use it to size layouts, typography and rectangles that feel balanced, and it shows up in the geometry of pentagons and pentagrams. Many claims of it appearing in nature and art are debated, so treat it as a useful design tool rather than a universal law.
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This calculator is for educational purposes. Double-check important results before acting on them.