Golden Ratio Calculator Formula

Understand the math behind the golden ratio calculator. Each variable explained with a worked example.

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Formulas Used

B

b = a / ((1 + sqrt(5)) / 2)

Total

total = a + a / ((1 + sqrt(5)) / 2)

Phi

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

Phi Sq

phi_sq = pow((1 + sqrt(5)) / 2, 2)

Variables

VariableDescriptionDefault
aLength a (longer segment)10

How It Works

Golden Ratio

Definition

phi = (1 + sqrt(5)) / 2 ≈ 1.6180339887...

Two quantities a and b (a > b > 0) are in the golden ratio if:

(a + b) / a = a / b = phi

Properties

  • phi² = phi + 1 ≈ 2.618
  • 1/phi = phi - 1 ≈ 0.618
  • Fibonacci ratio converges to phi

    • In Art and Nature

    The golden ratio appears in the Parthenon, Leonardo da Vinci's works, spiral shells, sunflower seeds, and DNA helices.

    Worked Example

    If the longer segment is 10, find the shorter segment.

    a = 10
    1. 01b = a / phi = 10 / 1.61803 ≈ 6.1803
    2. 02Total = 10 + 6.1803 = 16.1803
    3. 03Check: 16.1803 / 10 ≈ 1.618 = phi
    4. 04Check: 10 / 6.1803 ≈ 1.618 = phi

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