# What is the golden ratio split?

Splits a length in the golden ratio φ ≈ 1.618 from the whole, the longer part, or the shorter part, and gives the other two lengths.

- Page: https://www.acalculator.org/math/golden-ratio-calculator
- JSON spec: https://www.acalculator.org/math/golden-ratio-calculator.json
- Version: 022d5cd78628

## Default answer

Example with the default inputs (I know Whole (a + b), Length 66): The longer part is 40.7902 and the shorter part is 25.2098, for a whole of 66.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| k | I know | Which of the three lengths you know. |
| x | Length | The length you know, in any unit. The answer is in the same unit. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| long | Longer part a | The longer part: whole ÷ φ, or shorter × φ. |
| short | Shorter part b | The shorter part: longer ÷ φ, or whole − longer. |
| whole | Whole a + b | The longer part plus the shorter part. |
| phi | Golden ratio φ | (1 + √5) ÷ 2. |

## Method

φ = (1 + √5) ÷ 2 ≈ 1.6180339887. From the whole s: a = s ÷ φ, b = s − a. From the longer part a: b = a ÷ φ. From the shorter part b: a = b × φ. The whole is a + b.

## Assumptions

- The parts are in the golden ratio when (a + b) ÷ a = a ÷ b = φ.
- Lengths have no unit on the page: type them in any unit and read the answer in the same unit.
- φ is irrational, so lengths are computed in 64-bit floating point and rounded for display.

## Worked examples

1. k = whole, x = 66 gives long = 40.790243, short = 25.209757, phi = 1.618034. Source: OpenStax, Contemporary Mathematics, §13.1 Math and Art (φ = (1 + √5)/2 ≈ 1.618), https://openstax.org/books/contemporary-mathematics/pages/13-1-math-and-art (Example 13.1: 66 ÷ L = 1.618, L ≈ 40.8 in).
2. k = short, x = 10 gives long = 16.18034, whole = 26.18034. Source: OpenStax, Contemporary Mathematics, §13.1 Math and Art (φ = (1 + √5)/2 ≈ 1.618), https://openstax.org/books/contemporary-mathematics/pages/13-1-math-and-art.
3. k = long, x = 10 gives short = 6.18034, whole = 16.18034. Source: OpenStax, Contemporary Mathematics, §13.1 Math and Art (φ = (1 + √5)/2 ≈ 1.618), https://openstax.org/books/contemporary-mathematics/pages/13-1-math-and-art.

## FAQ

### What is the golden ratio?

The number φ = (1 + √5) ÷ 2 ≈ 1.6180339887. Two parts a (longer) and b (shorter) are in the golden ratio when (a + b) ÷ a = a ÷ b = φ.

### How do I split a length in the golden ratio?

Divide the whole by φ to get the longer part, and subtract it from the whole to get the shorter part. A 66 in length splits into 66 ÷ 1.618 = 40.79 in and 25.21 in.

### How do I find the shorter part from the longer one?

Divide by φ, or multiply by 0.618 (φ − 1 = 1 ÷ φ). A longer part of 10 has a shorter part of 6.18, and the whole is 16.18.

### What is a golden rectangle?

A rectangle whose long side ÷ short side is φ. With a short side of 10, the long side is 16.18. An 8 × 6 frame has a ratio of 1.333, so it is not a golden rectangle.

### How is the golden ratio linked to the Fibonacci numbers?

The ratio of two adjacent Fibonacci numbers gets closer to φ as the numbers grow: 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, and 75,025 ÷ 46,368 = 1.6180340.

### Which unit should I use?

Any unit. The page has no unit: type inches, centimetres or pixels, and read the answer in the same unit.

## Sources

- OpenStax, Contemporary Mathematics, §13.1 Math and Art (φ = (1 + √5)/2 ≈ 1.618; Examples 13.1, 13.3 and 13.4). https://openstax.org/books/contemporary-mathematics/pages/13-1-math-and-art (retrieved 2026-10-02)
- OpenStax, Calculus Volume 2, §5.1 Sequences (the Fibonacci ratios converge to φ = (1 + √5)/2). https://openstax.org/books/calculus-volume-2/pages/5-1-sequences (retrieved 2026-10-02)