In a golden rectangle, cutting off a square leaves a smaller golden rectangle, and repeating the process traces the golden spiral. The proportion is used in design, architecture and photography as a pleasing layout ratio.
How to use the Golden Rectangle Calculator
- Long side a
- Every result field is editable: type into Short side b or any other result and Long side a plus every other field update to match.
Results update as you type, and everything is calculated in your browser.
How it works
Enter the long side a or type any other value: b = a/φ, a + b = aφ and the area is a·b = a²/φ.
What each result means
- Short side b = [Long side a]÷((1 + sqrt(5))÷2)
- Long side + short side (a + b) = [Long side a]×((1 + sqrt(5))÷2)
- Area = [Long side a]^2÷((1 + sqrt(5))÷2)
- Perimeter = 2×([Long side a] + [Long side a]÷((1 + sqrt(5))÷2))
- Diagonal = sqrt([Long side a]^2 + ([Long side a]÷((1 + sqrt(5))÷2))^2)
Example values
How the main result changes when Long side a changes and everything else stays at the example values above:
| Long side a (cm) | Short side b (cm) |
|---|---|
| 5 | 3.0902 |
| 7.5 | 4.6353 |
| 10 | 6.1803 |
| 12.5 | 7.7254 |
| 15 | 9.2705 |
| 20 | 12.3607 |
Formula
a / b = (a + b) / a = φ = (1 + √5)/2 ≈ 1.6180339887
Worked example
Inputs — Long side a: 10 cm.
Results — Short side b = 6.18034 cm; Long side + short side (a + b) = 16.18034 cm; Area = 61.803399 cm²; Perimeter = 32.36068 cm; Diagonal = 11.755705 cm.
Key terms
- Golden ratio
- φ, the number with φ = 1 + 1/φ.
Frequently asked questions
What is the golden ratio exactly?
φ = (1 + √5)/2 ≈ 1.6180339887.
Are credit cards golden rectangles?
Nearly – 85.6 × 53.98 mm is about 1.586.
What happens when I cut a square from a golden rectangle?
The leftover rectangle is golden again.
How is it related to Fibonacci numbers?
Ratios of consecutive Fibonacci numbers approach φ.