Golden Ratio Calculator

Explore the golden ratio (phi = 1.618...). Check if a ratio is golden, calculate golden sections, and see the Fibonacci connection.

Golden Ratio Tools

Enter a value to find its golden section partner.

Result

Golden Ratio (phi)
1.61803399
phi (exact) (1 + sqrt(5)) / 2
phi decimal 1.61803398875...
1/phi (reciprocal) 0.61803398875...
phi squared 2.61803398875...

Step-by-Step Solution

phi = (1 + sqrt(5)) / 2 = 1.61803398875...

What Is the Golden Ratio?

The golden ratio, often denoted by the Greek letter phi, is an irrational mathematical constant approximately equal to 1.6180339887. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Algebraically, for a > b > 0, a/b = (a+b)/a = phi.

Key Properties of Phi

Algebraic Definition

Phi is the positive root of x² - x - 1 = 0.

phi = (1 + sqrt(5)) / 2

Reciprocal Property

The reciprocal of phi equals phi minus one.

1/phi = phi - 1 = 0.618...

Square Property

The square of phi equals phi plus one.

phi^2 = phi + 1 = 2.618...

Fibonacci Connection

Consecutive Fibonacci ratios converge to phi.

F(n+1)/F(n) -> phi as n -> inf

The Fibonacci Connection

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...) has a deep connection to the golden ratio. As you take the ratio of consecutive Fibonacci numbers, the values alternate above and below phi, getting closer with each step: 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, and so on, converging to phi.

Golden Ratio in Nature and Design

The golden ratio appears throughout nature and has been used in art and architecture for centuries:

  • Nature: Spiral patterns in sunflowers, pinecones, and nautilus shells often follow golden ratio proportions.
  • Art: Leonardo da Vinci, Salvador Dali, and other artists incorporated golden proportions in their works.
  • Architecture: The Parthenon, Notre-Dame, and many modern buildings use golden ratio proportions.
  • Design: Logo designers, photographers, and web designers use the golden ratio for aesthetic compositions.
  • Finance: Fibonacci retracement levels in technical analysis are based on the golden ratio.

Continued Fraction

Phi has the simplest possible continued fraction expansion: phi = 1 + 1/(1 + 1/(1 + 1/(1 + ...))). This makes it the "most irrational" number, in the sense that it is the hardest to approximate with rational numbers.

Golden Angle

The golden angle is approximately 137.508 degrees. It is the angle subtended by the smaller arc when a circle is divided into two arcs in the golden ratio. Plants often use this angle for leaf and seed arrangement to maximize sunlight exposure.