Golden Ratio Formula
The golden ratio phi = (1 + sqrt(5)) / 2 appears in geometry, art, and nature.
Learn the formula with worked examples.
The Formula
The golden ratio (phi) is an irrational number that appears throughout mathematics, art, architecture, and nature. Two quantities are in the golden ratio if their ratio equals the ratio of their sum to the larger quantity.
The golden ratio is closely related to the Fibonacci sequence. As Fibonacci numbers grow larger, the ratio of consecutive terms approaches phi.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| phi (golden ratio) | The golden ratio constant | dimensionless |
| a | The longer segment | any length unit |
| b | The shorter segment | same as a |
The Golden Ratio Property
If b = 1, then a = phi = 1.618...
Example 1 — Golden Rectangle
A golden rectangle has a short side of 10 cm. Find the long side.
Long side = short side x phi = 10 x 1.6180
Long side = 16.18 cm
Example 2 — Dividing a Line Segment
Divide a 50 cm line segment into two parts in the golden ratio.
Longer part = total / phi = 50 / 1.6180 = 30.90 cm
Shorter part = 50 - 30.90 = 19.10 cm
Check: 30.90 / 19.10 = 1.618 (confirms the golden ratio)
Longer part = 30.90 cm, Shorter part = 19.10 cm
Example 3 — Fibonacci Approximation
Show that consecutive Fibonacci numbers approximate the golden ratio.
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
8 / 5 = 1.600
13 / 8 = 1.625
21 / 13 = 1.6154
34 / 21 = 1.6190
55 / 34 = 1.6176
The ratios converge toward phi = 1.6180...
When to Use It
The golden ratio appears in many contexts:
- Art and design — creating aesthetically pleasing proportions
- Architecture — the Parthenon and many classical buildings use golden proportions
- Nature — spiral patterns in sunflowers, pinecones, and nautilus shells
- Typography — choosing font sizes and line spacing
- Photography — the golden spiral as a composition guide