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φ

Golden Rectangle

A golden rectangle has length/width = φ ≈ 1.618. Removing a square leaves another golden rectangle (self-similar). The Parthenon and credit cards use this proportion.

Concept Fundamentals
l/w = φ ≈ 1.618
Ratio
w = l/φ
Width
l = w × φ
Length
Remove square → golden
Self-similar

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Removing a square from a golden rectangle leaves another golden rectangle. φ = (1+√5)/2 is irrational; Fibonacci ratios approximate it. Credit cards (85.6×53.98 mm) approximate the golden ratio.

Key quantities
l/w = φ ≈ 1.618
Ratio
Key relation
w = l/φ
Width
Key relation
l = w × φ
Length
Key relation
Remove square → golden
Self-similar
Key relation

Ready to run the numbers?

Why: Golden rectangles appear in the Parthenon, Mona Lisa, credit cards, and iPhone screens. The proportion is perceived as aesthetically balanced.

How: Given one dimension: w = l/φ or l = w×φ. φ = (1+√5)/2 ≈ 1.618. Area = l×w = w²φ.

Removing a square from a golden rectangle leaves another golden rectangle.φ = (1+√5)/2 is irrational; Fibonacci ratios approximate it.
Sources:Wolfram MathWorld - Golden RatioKhan Academy - Geometry

Run the calculator when you are ready.

Golden Rectangle CalculatorEnter long side, short side, or area to compute dimensions
φ
GOLDEN RATIOφ ≈ 1.618

Golden Rectangle — Divine Proportion

Calculate dimensions from long side, short side, or area. Self-similar, aesthetically pleasing.

Rectangle →Square →

📐 Examples — Click to Load

Calculation Settings

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

φ

Golden rectangle: l/w = φ ≈ 1.618 — divine proportion.

— Definition

🏛️

The Parthenon façade fits a golden rectangle.

— Architecture

📋 Key Takeaways

  • • A golden rectangle has length/width = φ ≈ 1.618 — the divine proportion
  • • Removing a square from one side leaves another golden rectangle (self-similar)
  • • The Parthenon, Mona Lisa, and credit cards use golden proportions
  • • φ = (1+√5)/2 is irrational; Fibonacci ratios (8/5, 13/8) approximate it
  • • Given one dimension, all others follow: w = l/φ or l = w×φ

💡 Did You Know?

🏛️The Parthenon façade fits a golden rectangle — ancient Greeks valued φ in architectureSource: Architectural History
📱Credit cards (85.6×53.98 mm) approximate the golden ratio for pleasing proportionsSource: ISO 7810
🎨Leonardo da Vinci used golden rectangles in the Mona Lisa and Vitruvian ManSource: Art History
🌻Sunflower seed spirals follow Fibonacci numbers that converge to φSource: Phyllotaxis
📐The golden spiral is formed by quarter-circles in nested golden rectanglesSource: Wolfram MathWorld
🧬DNA double helix has a 34×21 nm pitch — close to the golden ratioSource: Molecular Biology

📖 Golden Rectangle Formulas

Golden Ratio (φ)

ϕ=1+52≈1.6180339887\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887

Length/width = φ. Width = length/φ. Length = width × φ.

From Area

w=Aϕ,l=w⋅ϕw = \sqrt{\frac{A}{\phi}}, \quad l = w \cdot \phi

🎯 Expert Tips

Design Harmony

Use golden rectangles for logos, frames, and layouts. The 1.618 ratio is perceived as balanced.

Quick Check

Measure length and width. If length/width ≈ 1.618, it's a golden rectangle.

Fibonacci Shortcut

Use consecutive Fibonacci numbers: 8×5, 13×8, 21×13 for quick approximations.

Area Mode

When you have a fixed area (e.g., 100 m²), use area mode to get golden proportions.

⚖️ Comparison Table

FeatureThis CalculatorBasicManual
Golden ratio modes (long/short/area)✅❌❌
Step-by-step solutions✅❌✅
Charts & visualization✅❌❌
Copy & share✅❌❌
7 real-world examples✅❌❌

📊 Quick Facts

1.618
φ (Golden Ratio)
φ:1
Length : Width
Self
Self-Similar
√5
In φ Formula

❓ FAQ

What is the golden ratio?

φ = (1+√5)/2 ≈ 1.618. It's the ratio where (a+b)/a = a/b. In a golden rectangle, length/width = φ.

How do I get width from length?

Divide length by φ: w = l/φ. Example: length 10 m → width = 10/1.618 ≈ 6.18 m.

Why do credit cards use golden ratio?

Standard cards are ~85.6×53.98 mm. 85.6/53.98 ≈ 1.586, close to φ for aesthetic appeal.

What is self-similarity?

Remove a square (side = width) from a golden rectangle. The remaining rectangle is also golden.

How does Fibonacci relate?

Ratios of consecutive Fibonacci numbers (8/5, 13/8, 21/13) approach φ.

Can I use this for art?

Yes. Many artists use golden rectangles for composition. Enter your canvas short side to get the long side.

What if I know the area?

Use "From Area" mode. Width = √(A/φ), then length = width × φ.

Is φ exact?

No. φ is irrational. We use the exact formula (1+√5)/2 in calculations.

⚠️ Disclaimer: This calculator uses the mathematical golden ratio φ = (1+√5)/2. Real-world objects (Parthenon, credit cards) may approximate but not exactly match these proportions due to historical and practical constraints.

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