Golden Ratio Calculator – φ = 1.6180339887

The golden ratio divides a line so the whole is to the large part as the large is to the small — the only ratio that equals its own square minus one: φ² = φ + 1. Numerically φ = (1+√5)/2 ≈ 1.6180339887, with reciprocal 1/φ ≈ 0.6180339887 (exactly φ − 1). The calculator splits any length: a 10 cm stick cuts into 6.18 cm and 3.82 cm, and the rectangle 6.18 × 3.82 is golden because 10/6.18 = 6.18/3.82 ≈ 1.618. Fibonacci ratios march toward φ — 13/8 = 1.625, 6765/4181 = 1.61803 — and the calculator shows your chosen Fibonacci pair converging. Powers follow the Lucas sequence (φ¹⁰ ≈ 122.99), and design presets mark the 61.8% and 38.2% levels used in layout grids and trading retracements.

1. 1

   Enter a length or ratio

   A total like 10 cm, or one segment to find the other

2. 2

   Divide by φ

   The larger part is total/φ = 6.18 cm; the smaller is the rest

3. 3

   Verify the proportion

   10/6.18 = 6.18/3.82 ≈ 1.618 is displayed

4. 4

   Explore Fibonacci links

   Consecutive ratios approach φ from both sides

• 1

  φ is the only number whose reciprocal equals itself minus 1

• 2

  φ² = φ + 1 = 2.618… — powers stay in the φ family

• 3

  Fibonacci neighbors bracket φ: 13/8 = 1.625, 21/13 ≈ 1.6154

• Rounding φ to 1.62 and losing the 0.0019 that matters in splits

• Confusing the golden ratio with π ≈ 3.14159 or e ≈ 2.71828

• Cutting at 61.8% from the wrong end — the larger part must be measured from the same reference

• Believing every spiral in nature is exactly golden; many only approximate it