Golden Ratio Calculator (φ = 1.618...)
The golden ratio (φ ≈ 1.6180339887) is a special mathematical ratio where (a+b)/a = a/b = φ. It appears in nature, art, architecture, and mathematics. A rectangle with side ratio φ:1 is a "golden rectangle," considered aesthetically pleasing.
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Tip: For designers: the golden ratio can guide layout proportions, type scale, and spacing. Many popular design systems (including some used by Apple and Google) incorporate golden ratio proportions.
- 1φ = (1 + √5) / 2 ≈ 1.6180339887
- 2Each Fibonacci number / previous Fibonacci number → φ as numbers grow larger
- 3Golden rectangle: if you remove a square from it, the remaining rectangle is another golden rectangle
- 4φ² = φ + 1 and 1/φ = φ − 1
Fibonacci sequence=1, 1, 2, 3, 5, 8, 13, 21, 34...21/13 = 1.615, 34/21 = 1.619, approaches φ
Rectangle with width 100=Golden height = 161.8100 × φ = 161.8
| Example | Relationship to φ |
|---|---|
| Nautilus shell | Spiral growth ratio ≈ φ |
| Sunflower seeds | 34 and 55 spirals (Fibonacci) |
| Human proportions | Many ratios approximate φ |
| Parthenon facade | Width/height ≈ φ |
| DNA structure | Width:height of full turn = φ |
| Pentagon diagonals | Diagonal/side = φ exactly |
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Fun Fact
The claim that the human body is full of golden ratios is largely myth — studies show many reported golden ratios in nature and the body are approximate at best. The real mathematical occurrences (pentagons, Fibonacci sequences) are genuine and remarkable.
References
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