What is the golden ratio split?
Type one length and say which one it is. The golden ratio calculator gives the other two, so that the whole over the longer part equals the longer part over the shorter part, both φ ≈ 1.618.
- Longer part a
- 40.7902
The longer part is 40.7902 and the shorter part is 25.2098, for a whole of 66.
- Shorter part b
- 25.2098
- Whole a + b
- 66
- Golden ratio φ
- 1.6180339887
Longer part a: 40.7902. The longer part is 40.7902 and the shorter part is 25.2098, for a whole of 66.
How to calculate
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.
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.
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.
- 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
Each example is checked against the calculator on every build.
- I know Whole (a + b), Length 66 gives Longer part a 40.790243, Shorter part b 25.209757, Golden ratio φ 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)
- I know Shorter part b, Length 10 gives Longer part a 16.18034, Whole a + b 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
- I know Longer part a, Length 10 gives Shorter part b 6.18034, Whole a + b 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
How it works
The golden ratio is φ = (1 + √5) ÷ 2 = 1.6180339887…. A whole split into a longer part a and a shorter part b is in the golden ratio when (a + b) ÷ a = a ÷ b = φ.
- From the whole s: a = s ÷ φ, then b = s − a.
- From the longer part a: b = a ÷ φ, and the whole is a + b.
- From the shorter part b: a = b × φ, and the whole is a + b.
Rules
- The length you type is more than 0 and at most 10¹². It has no unit; the answer is in the unit you typed.
- φ is irrational, so the lengths are computed in 64-bit floating point.
Output format. The three lengths show with at most 4 decimals and φ with 10, each rounded half up from the computed value.
Worked examples by hand
A height of 66 in (OpenStax Example 13.1). a = 66 ÷ 1.6180339887 = 40.7902; b = 66 − 40.7902 = 25.2098. The book rounds φ to 1.618 and gets 40.8 in.
A shorter part of 10. a = 10 × 1.6180339887 = 16.1803; whole = 16.1803 + 10 = 26.1803.
A longer part of 10. b = 10 ÷ 1.6180339887 = 6.1803; whole = 16.1803.
Other questions people ask
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.