PhiLab

Golden Ratio Calculator

φ = (1 + √5) / 2 ≈ 1.6180339887.

Value a

Golden ratio × a

1.618

φ ≈ 1.6180339887…

φ (constant): 1.618

What you entered

Value a: 1

1
The golden ratio constant φ
(1 + √5) ÷ 2= 1.618
2
Multiply by a
1 × 1.618= 1.618

Result

Golden ratio × a: 1.618

1 × φ = 1.618.

Golden ratio partners for common lengths

Multiplying a length by phi, about 1.618034, gives the longer side of a golden rectangle.

Length	Times phi	Phi
1	1.6180	1.618034
5	8.0902	1.618034
10	16.1803	1.618034
16	25.8885	1.618034
100	161.8034	1.618034
1920	3106.6253	1.618034

Phi is the only number where subtracting 1 gives its own reciprocal: 1.618034 minus 1 is 0.618034, which is 1 divided by 1.618034. That self-similarity is why removing a square from a golden rectangle leaves a smaller golden rectangle, and why the proportion recurs when you subdivide. Its exact value is (1 + the square root of 5) divided by 2, so it is irrational. Claims that it governs classical architecture and Renaissance painting are mostly retrofitted - the reliable appearances are in the Fibonacci sequence and in phyllotaxis, the arrangement of leaves and seeds.

A ratio that reproduces itself

φ is the unique number where the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part — a self-similar proportion that repeats at every scale.

Where designers use it

Artists and architects have used golden-ratio proportions for centuries in composition and layout because rectangles built to this ratio are widely perceived as visually balanced.

Frequently asked questions

What is the golden ratio?

The golden ratio φ = (1 + √5) / 2 ≈ 1.6180339887. It's the ratio where a/b = (a+b)/a. A golden rectangle (sides in ratio 1:φ) can be divided into a square and a smaller golden rectangle, repeating infinitely.

Where does the golden ratio appear?

Art and architecture (Parthenon proportions, Le Corbusier's designs), nature (Fibonacci spirals, sunflower seeds, shell curves), typography (font sizing ratios), and finance (Fibonacci retracement levels in stock analysis).

How is the golden ratio related to Fibonacci numbers?

The ratio of consecutive Fibonacci numbers converges to φ. F(n+1)/F(n) → φ as n → ∞. Examples: 5/3 = 1.667, 8/5 = 1.600, 13/8 = 1.625, 21/13 = 1.615, 34/21 = 1.619. The convergence is rapid.