About

The golden ratio φ = 1 + √52 ≈ 1.6180339887 is an irrational constant that appears in geometry, architecture, financial analysis, and biological growth patterns. A line segment divided such that the ratio of the whole a + b to the longer part a equals the ratio of a to the shorter part b satisfies this proportion exactly. Errors in applying φ lead to visually off compositions in design, structurally suboptimal proportions in engineering, and incorrect Fibonacci-based retracement levels in trading. This calculator takes any known segment length and computes all related golden-ratio quantities: the complementary segment, the total length, the golden rectangle dimensions, and the corresponding Fibonacci neighborhood. It also renders the golden spiral on canvas so you can verify proportional harmony visually. The tool assumes Euclidean geometry and positive real inputs. Precision is maintained to 10 decimal places internally, displayed to 6.

Formulas

The defining proportion of the golden ratio:

a + ba = ab = φ

Closed-form value:

φ = 1 + √52

Given a (longer segment), the shorter segment:

b = aφ

Given b (shorter segment), the longer segment:

a = b × φ

Given total a + b, the longer segment:

a = (a + b)φ

Binet's formula for the n-th Fibonacci number:

F_n = φⁿ − ψⁿ√5

Where ψ = 1 − √52 ≈ −0.6180339887 is the conjugate root.

Variable legend: a = longer segment. b = shorter segment. φ = golden ratio (1.618…). ψ = conjugate of φ. F_n = n-th Fibonacci number.

Reference Data

Property	Value / Relationship
Golden Ratio φ	1.6180339887…
Reciprocal 1/φ	0.6180339887… = φ − 1
φ²	2.6180339887… = φ + 1
φ³	4.2360679775… = 2φ + 1
φ⁴	6.8541019662… = 3φ + 2
φ⁵	11.0901699437… = 5φ + 3
Continued Fraction	1 + 11 + 11 + …
Fibonacci Ratio Limit	lim F_n+1 / F_n = φ
Decimal Expansion (50 digits)	1.61803398874989484820458683436563811772030917980576
Golden Angle	137.5077° ≈ 360φ²
Pentagon Diagonal / Side	φ
Icosahedron Edge Ratio	Involves φ in vertex coordinates
Lucas Numbers Ratio Limit	lim L_n+1 / L_n = φ
φ in Quadratic	x² − x − 1 = 0
Negative Root	−0.6180339887… = −1/φ
Fibonacci 1-20	1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765
F₃₀	832040
F₄₀	102334155
F₅₀	12586269025
Silver Ratio δ_S	2.4142135624… (for comparison)
Plastic Number	1.3247179572… (for comparison)

Frequently Asked Questions

The Fibonacci recurrence F(n+1) = F(n) + F(n−1) implies F(n+1)/F(n) = 1 + F(n−1)/F(n). As n → ∞, both ratios approach the same limit L, so L = 1 + 1/L, yielding L² − L − 1 = 0. The positive root is φ. Convergence is geometric, with error proportional to (−1/φ)^n, so by F(12)/F(11) = 144/89 ≈ 1.61798, you already have 4-digit accuracy.

JavaScript uses IEEE 754 double-precision floats with ~15-17 significant digits. For inputs below 10⁻¹⁰ or above 10¹⁵, multiplication by φ can lose trailing digits. This tool caps display at 6 decimal places and internally uses the full constant 1.6180339887498948482 (18 digits), which stays within float64 safe range. For cryptographic or astronomical precision, arbitrary-precision libraries would be needed.

The Parthenon claim is disputed. Measured façade ratios vary between 1.71 and 1.75 depending on which features you measure, which deviates from φ by 6-8%. The nautilus shell follows a logarithmic spiral, but its growth factor is typically around 1.33 per quarter turn, not φ. True golden spirals appear in phyllotaxis (sunflower seed packing) because the golden angle (≈137.508°) maximizes packing efficiency, a result proven by Douady and Couder in 1992.

In a regular pentagon with side length s, each diagonal has length s × φ. Furthermore, diagonals intersect each other in golden ratio proportions: each diagonal is divided by the other into segments of ratio φ:1. This is the geometric basis for constructing φ with straightedge and compass. The interior angle of 108° and the isosceles triangle with angles 72° - 72° - 36° (the 'golden gnomon') are both intimately linked to φ.

Traders use ratios derived from φ - specifically 23.6% (1/φ⁴), 38.2% (1/φ²), 50%, 61.8% (1/φ), and 78.6% (√(1/φ)) - as potential support and resistance levels after a price move. These are applied by measuring a swing high-to-low and marking horizontal lines at those percentages. Empirical evidence for their predictive power is weak; academic studies (e.g., Malkiel's 'A Random Walk Down Wall Street') classify them as self-fulfilling only when enough traders act on them simultaneously.

The quadratic x² − x − 1 = 0 has two real roots: φ ≈ 1.618 and ψ ≈ −0.618. The negative root ψ = −1/φ = 1 − φ appears in Binet's formula and in the theory of continued fractions. There is no complex solution because the discriminant (1 + 4 = 5) is positive. However, generalizations to metallic means (solutions of x² − nx − 1 = 0) and complex-valued analogs exist in algebraic number theory.