Golden Ratio Calculator Formula
Understand the math behind the golden ratio calculator. Each variable explained with a worked example.
Formulas Used
B
b = a / ((1 + sqrt(5)) / 2)Total
total = a + a / ((1 + sqrt(5)) / 2)Phi
phi = (1 + sqrt(5)) / 2Phi Sq
phi_sq = pow((1 + sqrt(5)) / 2, 2)Variables
| Variable | Description | Default |
|---|---|---|
a | Length 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
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.
- 01b = a / phi = 10 / 1.61803 ≈ 6.1803
- 02Total = 10 + 6.1803 = 16.1803
- 03Check: 16.1803 / 10 ≈ 1.618 = phi
- 04Check: 10 / 6.1803 ≈ 1.618 = phi
Frequently Asked Questions
What is the golden ratio?
The golden ratio (phi ≈ 1.618) is an irrational number that appears when a line is divided so that the ratio of the whole to the larger part equals the ratio of the larger part to the smaller.
Where does the golden ratio appear in nature?
In spiral shells (nautilus), sunflower seed patterns, pine cones, the arrangement of leaves on stems, and the proportions of the human body.
How is the golden ratio related to Fibonacci numbers?
The ratio of consecutive Fibonacci numbers converges to the golden ratio. F(n+1)/F(n) → phi as n increases.
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