Golden Ratio Calculator
Calculate all three values of a golden ratio division — short side a, long side b, and total length a+b — given any one of them. φ ≈ 1.6180339887.
Did this tool work for you?
How to use this calculator
The golden ratio φ divides a line so that the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part.
- 1
Select which dimension you already know: short side a, long side b, or the total length.
- 2
Enter its value.
- 3
The other two dimensions are calculated using the golden ratio φ ≈ 1.618.
Frequently asked questions
What is the golden ratio?
The golden ratio φ = (1+√5)/2 ≈ 1.6180339... is a mathematical constant defined by the property that the ratio of the sum (a+b) to the larger part (b) equals the ratio of the larger to the smaller (b/a). Both equal φ. It is irrational and appears in geometry, art, architecture, and nature.
How does the golden ratio relate to Fibonacci numbers?
Consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, …) converge to the golden ratio: 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.6154, 55/34 ≈ 1.6176, approaching φ ≈ 1.61803. The exact relationship: φ = (1+√5)/2 is the solution to φ² = φ+1, the same recurrence that defines Fibonacci numbers.
Where does the golden ratio appear in nature?
Phyllotaxis (the arrangement of leaves, seeds, and petals in plants) often uses Fibonacci numbers of spirals — sunflower seeds are arranged in 34 and 55 counter-rotating spirals, consecutive Fibonacci numbers. Shell spirals (nautilus), galaxy arm patterns, and DNA molecule dimensions have all been cited in popular science as golden-ratio examples, though some are approximations.
Is the golden ratio actually used in design and architecture?
The golden ratio is used as a design guideline in typography, logo design, UI layout, and photography composition. Claims about the Parthenon and the Mona Lisa fitting exact golden ratios are disputed; measurements vary by source. Regardless, rectangles with golden proportions are widely considered aesthetically pleasing, and the rule of thirds in photography is a practical approximation.
Golden ratio calculator — φ = 1.6180339887
Definition and algebraic properties
The golden ratio φ satisfies φ² = φ + 1, which rearranges to φ² − φ − 1 = 0. The positive root is φ = (1+√5)/2 ≈ 1.61803. Its reciprocal 1/φ = φ−1 ≈ 0.61803 — the only positive number whose reciprocal is exactly one less than itself. This self-similar property means φ has the continued fraction [1; 1, 1, 1, …] — the "slowest converging" continued fraction, making it the "most irrational" number in a precise mathematical sense.
Golden rectangles and spirals
A golden rectangle has sides in the ratio 1:φ. Removing a square from a golden rectangle leaves a smaller golden rectangle — a self-similar property that can be repeated infinitely. Connecting the corners of successive squares with quarter-circle arcs traces the golden spiral, a logarithmic spiral with constant growth factor φ per quarter turn. This spiral is frequently cited as appearing in nautilus shells and galaxy shapes.
Fibonacci connection and the Lucas sequence
The formula Fₙ = (φⁿ − ψⁿ)/√5 (Binet's formula, where ψ = −1/φ ≈ −0.618) gives the nth Fibonacci number exactly. This means Fibonacci numbers are encoded in powers of the golden ratio. The related Lucas numbers (2, 1, 3, 4, 7, 11, 18, 29, …) obey the same recurrence (Lₙ = Lₙ₋₁ + Lₙ₋₂) and also converge to φ in consecutive ratios.
Learn more from an authoritative source:
WikipediaScientific Calculator
Evaluate mathematical expressions including trigonometry, logarithms, exponents, and more.
Percentage Calculator
Quickly calculate percentages, percentage change, and percentage of a total.
Fraction Calculator
Add, subtract, multiply, or divide two fractions and get the simplified result instantly.
Ratio Calculator
Simplify ratios, find missing values, and scale ratios up or down.
Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →