Golden Ratio Calculator
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Two lengths a > b are in the golden ratio when (a + b)/a = a/b = φ = (1 + √5)/2 ≈ 1.618. To split 10 cm, the longer part is 10 ÷ φ ≈ 6.18 cm and the shorter part is ≈ 3.82 cm.
Any unit: cm, inches, pixels.
The green square (a by a) plus the orange golden rectangle (b by a) make a larger golden rectangle; the arcs start the golden spiral.
| Golden proportion | |
| Longer part a | 6.180339887 |
| Shorter part b | 3.819660113 |
| Total a + b | 10 |
| a as a share of the total | 61.80% |
| Golden rectangle | |
| Sides (a + b) by a | area 61.803399 |
| Facts about φ | |
| φ | 1.6180339887… |
| 1/φ = φ − 1 | 0.6180339887… |
| φ² = φ + 1 | 2.6180339887… |
| Fibonacci ratio 89/55 | 1.6181818… |
Show steps
- The golden ratio is \(\varphi = \frac{1+\sqrt{5}}{2} \approx 1.6180339887\).
- Golden proportion: \(\frac{a+b}{a} = \frac{a}{b} = \varphi\). You gave the total length a + b = 10.
- Longer part: a = total ÷ φ = 10 ÷ 1.6180339887 = 6.1803399 (about 61.8%). Shorter part: b = total − a = 3.8196601 (about 38.2%).
- Check: a ÷ b = 1.618033989 and (a + b) ÷ a = 1.618033989. Both equal φ.
How to use the golden ratio calculator
- Enter one length.
- Say whether it is the whole length, the longer part or the shorter part.
- Press Calculate to get the other two lengths and a golden rectangle drawing.
The golden ratio formula
From a/b = (a + b)/a, set φ = a/b. Then φ = 1 + 1/φ, so φ² = φ + 1. Solving the quadratic gives \(\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\).
| You know | Longer part a | Shorter part b |
|---|---|---|
| Total T | T ÷ φ ≈ 0.618T | T − a ≈ 0.382T |
| Longer part a | a | a ÷ φ |
| Shorter part b | b × φ | b |
Golden ratio and the Fibonacci numbers
Divide each Fibonacci number by the one before it: 3/2 = 1.5, 5/3 ≈ 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 34/21 ≈ 1.619. The ratios get closer and closer to φ.
Where the golden ratio appears
It appears in a regular pentagon (the diagonal ÷ the side is φ), in the golden rectangle and spiral, and in some plant spirals linked to Fibonacci numbers. Many popular claims about it, for example in famous paintings or the human body, are not supported by careful measurement.
Golden ratio practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
What is the golden ratio in simple terms?
A way to split a length so the whole compares to the longer part exactly as the longer part compares to the shorter part. That ratio is about 1.618.
How do I calculate the golden ratio of a number?
Multiply by 1.618 to get the next larger golden length, or multiply by 0.618 to get the next smaller one.
Is 16:9 the golden ratio?
No. 16:9 is 1.778; the golden ratio is 1.618. A golden rectangle is a bit squarer than a widescreen screen.
Why is φ called irrational?
It involves √5, which cannot be written as a fraction, so its decimal digits never end or repeat.
What is the golden ratio used for in design?
Some designers use it to choose proportions for layouts, logos or type sizes. It is one option among many, not a rule.
Sources
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