# Golden Rectangle Calculator

URL: https://speedycalculator.com/calculators/math/golden-rectangle/
Last updated: 2026-10-30

> The golden rectangle calculator finds the short side b, the sum a + b and the area of a rectangle whose sides are in the golden ratio φ ≈ 1.618 – or works backwards from any of them.

## Inputs

- Long side a (cm, m, mm, in, ft, yd, km, mi)

## Results

- Short side b
- Long side + short side (a + b)
- Area
- Perimeter
- Diagonal

## Formula

```
a / b = (a + b) / a = φ = (1 + √5)/2 ≈ 1.6180339887
```

## Explanation

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)53.09027.54.6353106.180312.57.7254159.27052012.3607

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Golden rectangle](https://mathworld.wolfram.com/GoldenRectangle.html)
- [Wolfram MathWorld – Golden ratio](https://mathworld.wolfram.com/GoldenRatio.html)