Golden Ratio Calculator | φ = 1.618033…
Calculate golden ratio dimensions from any measurement. Find A from B, B from A, total length, golden rectangle dimensions. Shows Fibonacci spiral convergence to φ with up to 15 decimal precision.
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What Is the Golden Ratio Calculator | φ = 1.618033…?
The golden ratio φ (phi) is the unique positive number satisfying x² = x + 1, giving φ = (1+√5)/2 ≈ 1.6180339887. It is irrational, its decimal expansion never repeats. A line divided in the golden ratio has the remarkable property that the whole is to the longer part as the longer part is to the shorter part.
- ▸B from A: Enter the longer dimension to find the shorter golden partner B = A/φ.
- ▸A from B: Enter the shorter dimension to find the longer golden partner A = B×φ.
- ▸A+B from A: Finds the total length that makes A the larger golden segment.
- ▸Fibonacci tab: Shows 20 Fibonacci terms with ratios converging to φ with precision.
- ▸Rectangle tab: Visualises the golden rectangle given any width.
Formula
Find B from A
B = A / φ
Given the longer segment A, the shorter segment B = A ÷ φ.
Find A from B
A = B × φ
Given the shorter segment B, the longer segment A = B × φ.
Find total (A+B)
Total = A × φ
The total length A+B equals A times φ.
Key identity
φ² = φ + 1 ≈ 2.618
The square of φ equals φ plus 1. Also: 1/φ = φ − 1 ≈ 0.618.
How to Use
- 1
Select a mode
Choose B from A, A from B, A+B from A, Fibonacci Spiral, or Rectangle Dimensions depending on what you want to compute.
- 2
Try a preset
Click Canvas 800px, A4 Paper, Instagram, Business Card, or iPhone to pre-fill a real design dimension.
- 3
Choose decimal precision
Select 4 to 15 decimal places for the output using the precision selector.
- 4
Enter your value
Type any positive measurement, pixels, mm, inches, or any unit.
- 5
Press Calculate
Results show A, B, total, and both ratios confirming the golden relationship.
- 6
View the visual
A proportional rectangle diagram shows the A:B split with your actual values labeled.
- 7
Explore Fibonacci
Switch to Fibonacci tab to see 20 Fibonacci numbers and how their ratios converge to φ.
- 8
Copy results
Use the Copy Results button to copy all values to your clipboard.
Example Calculation
Example 1 | Web canvas 800px wide
A sidebar of 494 px alongside a main content area of 800 px creates a golden layout.
Example 2 | A4 paper 297mm
A4 has a ratio of 297/210 ≈ 1.414, which is √2, not φ, a common misconception.
Example 3 | Fibonacci convergence
Each additional Fibonacci term adds roughly 0.7 digits of φ precision.
Understanding Golden Ratio | φ = 1.618033…
What Is the Golden Ratio?
The golden ratio φ (phi) is the positive solution to x² = x + 1. This simple equation yields the irrational number (1+√5)/2 ≈ 1.6180339887. It has the unique property that φ² = φ + 1 and 1/φ = φ − 1. All calculations run live in your browser.
Golden Ratio in Nature
- ▸Phyllotaxis: Sunflower seeds, pine cone spirals, and leaf arrangements use Fibonacci numbers, producing golden angle spirals.
- ▸Nautilus shell: Approximates a golden spiral, though not exactly φ.
- ▸DNA molecule: One full cycle is 34Å long and 21Å wide, consecutive Fibonacci numbers.
- ▸Galaxy arms: Logarithmic spirals in spiral galaxies approximate golden spirals.
Golden Ratio in Design and Architecture
- ▸Parthenon: Often cited as using φ, though precise measurements are debated.
- ▸Renaissance art: Da Vinci and others consciously applied golden proportions.
- ▸Typography: Golden ratio used for font size hierarchies (e.g., body 16px, heading 26px).
- ▸UI design: Column widths, card proportions, and spacing systems based on φ.
The Fibonacci Connection
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21…) is defined by F(n) = F(n−1) + F(n−2). As n grows, the ratio F(n+1)/F(n) converges to φ with exponential speed. By term 20, the approximation is accurate to 10 decimal places. This convergence can be proved using the closed-form Binet formula: F(n) = (φⁿ − ψⁿ)/√5, where ψ = (1−√5)/2.
Frequently Asked Questions
What is the golden ratio exactly?
The golden ratio φ is the positive real number satisfying x² = x + 1, giving:
- • φ = (1+√5)/2 ≈ 1.6180339887…
- • It is irrational and transcendental.
- • Key identities: φ² = φ+1, 1/φ = φ−1 ≈ 0.618
How is the golden ratio related to Fibonacci numbers?
Consecutive Fibonacci numbers converge to φ in their ratio:
- • F(n+1)/F(n) → φ as n → ∞
- • 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.6154, 55/34 ≈ 1.6176
- • Binet formula: F(n) = (φⁿ − ψⁿ)/√5
This connection is why golden spirals appear wherever Fibonacci spirals are found in nature.
What is a golden rectangle?
A golden rectangle has sides in ratio φ:1. Its defining property is self-similarity:
- • Remove a square from one end → the remaining piece is another golden rectangle
- • Repeating this produces the golden spiral
- • Given width W: height = W / φ ≈ 0.618 × W
Is the golden ratio really used in famous art and buildings?
Some appearances are real, many are exaggerated or retrofitted:
- • Real: Le Corbusier's Modulor system explicitly uses φ for human proportions
- • Debated: The Parthenon and Great Pyramid claims depend heavily on which measurements are chosen
- • Confirmed: Salvador Dalí's Sacrament of the Last Supper uses a golden rectangle frame deliberately
What is the golden angle?
The golden angle is 360° divided by φ², giving approximately 137.5°:
- • θ = 360° / φ² ≈ 137.508°
- • Plants use this angle between successive leaves to maximize sunlight exposure
- • It produces the Fibonacci spiral patterns in sunflowers and pinecones
Can I use the golden ratio for web design?
Yes, the golden ratio is actively used in modern UI/UX design:
- • Typography: If body text is 16px, heading ≈ 16×φ ≈ 26px
- • Column layout: Main column vs sidebar in ratio φ:1 (≈ 62%:38%)
- • Spacing: Margin/padding ratios following φ feel naturally harmonious
- • Use this calculator's presets (Canvas 800px, etc.) for instant design values
What is the reciprocal of the golden ratio?
The reciprocal 1/φ has an elegant value:
- • 1/φ = φ − 1 ≈ 0.6180339887…
- • Note the decimal part is identical to φ itself
- • This is unique among positive numbers: x − 1/x = 1 has only the golden ratio as solution
How many decimal places is this calculator accurate to?
This calculator uses JavaScript's 64-bit floating-point arithmetic (IEEE 754 double precision), which provides approximately 15–17 significant digits. The precision selector lets you display 4 to 15 decimal places. All calculations run instantly in your browser with no server needed.