Golden Ratio Calculator

What Is the Golden Ratio? The golden ratio (φ, phi) is a special mathematical constant approximately equal to 1.6180339887. It is the ratio of two quantities where the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part.

Definition If A > B, then (A + B) / A = A / B = φ ≈ 1.6180339887

Exact Value φ = (1 + √5) / 2 ≈ 1.6180339887… 1/φ = φ − 1 ≈ 0.6180339887…

Remarkably, 1/φ = φ − 1. No other number has this property.

Fibonacci Connection The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… As the sequence progresses, the ratio of consecutive terms (8/5, 13/8, 21/13…) converges to φ. By the 10th term, the ratio is accurate to 4 decimal places.

Golden Rectangle A golden rectangle has sides in the ratio 1:φ. If you cut a square from it, the remaining piece is also a golden rectangle — infinitely recursive.

Width W → Height H = W/φ (landscape) Width W → Height H = W × φ (portrait)

Golden Spiral A logarithmic spiral that grows by a factor of φ for every 90° turn. Found in nautilus shells, galaxy arms, and the arrangement of seeds in sunflowers.

Appearances in Nature

• Phyllotaxis: sunflower seeds and pine cone spirals follow Fibonacci counts (8 and 13, or 13 and 21)
• The human body: the ratio of forearm to hand, and naval to foot length, approximate φ in many people
• DNA: one full helix cycle is 34 Å long and 21 Å wide: two consecutive Fibonacci numbers

In Art and Architecture

• The Parthenon in Athens (447–432 BC): facade proportions closely approximate φ
• Leonardo da Vinci used the golden ratio in the Vitruvian Man (1490)
• Le Corbusier’s Modulor design system was based on the golden ratio

One more property worth knowing. φ has the simplest possible continued fraction expansion: [1; 1, 1, 1, 1, …] — every coefficient is just 1. That makes it, in a precise mathematical sense, the most irrational number — the hardest to approximate well with simple fractions. This is part of why φ shows up so often in nature (leaf arrangements, seed spirals): biological systems that need to pack things without overlap settle on the angle that resists periodic alignment, and that angle is 360° / φ² ≈ 137.5°.

A note on Fibonacci retracement in finance. Stock traders use Fibonacci ratios (23.6%, 38.2%, 50%, 61.8%, 78.6%) as “support and resistance” levels on price charts — the 61.8% level comes directly from 1/φ. Whether the math has any real predictive value is debated, but the practice is widespread enough that the levels become self-fulfilling: traders watch for them, place orders there, and the chart sometimes pivots there because of the orders, not because of φ.