Golden Ratio Calculator

Calculate golden ratio proportions with step-by-step solutions, Fibonacci connections, and geometric visualizations.

Formula:phi = (1 + sqrt(5)) / 2 = 1.618...

Result

Golden Ratio

1.6180339887

Segment A10
Segment B6.1803398875
Total (A+B)16.1803398875
A/B Ratio1.6180339887

Calculate Golden Proportions

Enter the known segment length

Which value do you know?

Number of decimal places (2-15)

Golden Ratio Division

A/B = 1.6180339887

A
B
A = 10B = 6.1803

Step-by-Step Solution

1

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

2

phi = 1.6180339887

3

Given: Segment A = 10

4

Using the golden ratio: A / B = phi

5

B = A / phi = 10 / 1.618034 = 6.1803398875

6

Total = A + B = 10 + 6.1803398875 = 16.1803398875

7

Verification:

8

A / B = 10 / 6.1803398875 = 1.6180339887

9

(A + B) / A = 16.1803398875 / 10 = 1.6180339888

A/B = 1.6180339887 (should be close to 1.6180339887)

phi squared

phi^2 = phi + 1

1.618034^2 = 2.618034 = 1.618034 + 1

Reciprocal

1/phi = phi - 1

1/1.618034 = 0.618034 = 1.618034 - 1

phi cubed

phi^3 = 2*phi + 1

1.618034^3 = 4.236068

Continued fraction

phi = 1 + 1/(1 + 1/(1 + ...))

The golden ratio can be expressed as an infinite continued fraction of all 1s.

Result

Golden Ratio

1.6180339887

Segment A10
Segment B6.1803
A/B Ratio1.618034

?How Do You Calculate the Golden Ratio?

The golden ratio (phi) = (1 + sqrt(5)) / 2 = 1.6180339887... When a line is divided in golden ratio, A/B = (A+B)/A = phi. To find golden proportions: if A is known, B = A/phi; if B is known, A = B x phi. The ratio appears in Fibonacci sequences (consecutive Fibonacci numbers approach phi), art, architecture, and nature.

What is the Golden Ratio?

The golden ratio (phi, or the Greek letter phi) is an irrational number approximately equal to 1.618. Two quantities are in the golden ratio if their ratio equals the ratio of their sum to the larger quantity: a/b = (a+b)/a = phi. It appears throughout nature, art, and architecture. The Fibonacci sequence is intimately connected, with consecutive ratios converging to phi.

Key Facts About the Golden Ratio

  • Golden ratio (phi) = (1 + sqrt(5)) / 2 = 1.6180339887...
  • A/B = phi means A and B are in golden proportion
  • phi^2 = phi + 1 (unique property)
  • 1/phi = phi - 1 = 0.6180339887...
  • Fibonacci ratios approach phi: 13/8 = 1.625, 21/13 = 1.615...
  • Golden rectangle: sides in ratio 1:phi
  • Found in Parthenon, Mona Lisa, spiral shells
  • Also called divine proportion or golden section

Frequently Asked Questions

The golden ratio (phi) is approximately 1.6180339887. When a line is divided at the golden ratio, the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part. Mathematically: (a+b)/a = a/b = phi.
The golden ratio phi = (1 + sqrt(5)) / 2. This comes from solving x^2 = x + 1, which gives x = (1 + sqrt(5))/2. The negative solution (1 - sqrt(5))/2 = -0.618... is called the golden ratio conjugate.
Consecutive Fibonacci numbers (1,1,2,3,5,8,13,21...) have ratios that approach phi. For example: 8/5=1.6, 13/8=1.625, 21/13=1.615..., converging to 1.618... This is because the Fibonacci recurrence F(n)=F(n-1)+F(n-2) mirrors the golden ratio equation.
The golden ratio appears in spiral shells (nautilus), sunflower seed arrangements (137.5 degree angle), leaf phyllotaxis, hurricane spirals, DNA molecule proportions, and the branching of trees. These patterns emerge because phi provides optimal packing and growth efficiency.
A golden rectangle has sides in the ratio 1:phi (approximately 1:1.618). When you remove a square from a golden rectangle, the remaining rectangle is also a golden rectangle. This self-similar property creates the logarithmic golden spiral.

Last updated: 2025-01-15