Golden Ratio Calculator (φ)
Split a length in the golden ratio or find one part from another: the total, the long part and the short part, at φ ≈ 1.618. For design, architecture and photography, with the Fibonacci numbers.
- Total length (a + b)100 cm
- Long part (a)61.8034 cm
- Short part (b)38.1966 cm
- Share of the long part (1 ÷ φ)61.8%
- Share of the short part (1 ÷ φ²)38.2%
- Check: a ÷ b = (a + b) ÷ a1.6180339887
- Golden ratio φ = (1 + √5) ÷ 21.6180339887
Fibonacci numbers closing in on φ
How it is calculated
The golden ratio divides a length so that the whole is to the long part as the long part is to the short one. The ratio is irrational: φ ≈ 1.6180339887. It is the only positive number with φ² = φ + 1 and 1 ÷ φ = φ − 1. Ratios of consecutive Fibonacci numbers close in on φ. Designers and photographers use it as a composition guide; examples in nature and art are often only approximate.
φ = (1 + √5) ÷ 2 ≈ 1.6180339887 · (a + b) ÷ a = a ÷ b = φa: long part · b: short part · a + b: total length. From the total, a = total ÷ φ and b = total − a; from the long part, b = a ÷ φ.- Pick the length you know and which part it is.
- From the total, the long part is total ÷ φ (61.8% of it) and the short part the rest (38.2%).
- From the long part, the short part is a ÷ φ and the total a × φ; from the short part, the long part is b × φ.
- Check: long ÷ short and total ÷ long give the same number, φ.
Example
100 cm splits into 61.8034 cm and 38.1966 cm. For a 210 cm door the long part is 210 ÷ 1.6180 ≈ 129.79 cm and the short part ≈ 80.21 cm. A 1,000 px page divides into 618 px of content and a 382 px sidebar.
Frequently asked questions
Why is the golden ratio 1.618?
Writing the definition (a + b) ÷ a = a ÷ b = φ gives the equation φ² = φ + 1, whose positive root is (1 + √5) ÷ 2 = 1.6180339887….
How is it related to the Fibonacci numbers?
In the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …) the ratio of two consecutive numbers gets ever closer to φ: 13 ÷ 8 = 1.625, 21 ÷ 13 ≈ 1.6154, 89 ÷ 55 ≈ 1.6182. The result table lists the first 20 ratios and their distance from φ.
How do I draw a golden rectangle?
Enter the long side as the “Long part”; the short side is the short part in the result. A golden rectangle with a 30 cm long side has a short side of 30 ÷ 1.618 ≈ 18.54 cm.