What the golden ratio is
Two lengths are in the golden ratio when the whole relates to the longer part exactly as the longer part relates to the shorter one: (a + b) ÷ a = a ÷ b. The only positive number that satisfies this is φ (phi) = (1 + √5) ÷ 2 ≈ 1.6180339887. It appears in the regular pentagon, in the Fibonacci sequence and in many design grids, and it has a famous reputation in art and architecture, some of it deserved and some of it myth.
The formulas
- From the whole: a = total ÷ φ and b = a ÷ φ, so the parts are about 61.8% and 38.2% of the total.
- From the longer part: b = a ÷ φ and total = a × φ.
- From the shorter part: a = b × φ.
Because φ² = φ + 1, the numbers behave neatly: 1 ÷ φ = φ − 1 ≈ 0.618.
Worked example: splitting 100 cm
- Longer part: 100 ÷ 1.618 = 61.80 cm.
- Shorter part: 61.80 ÷ 1.618 = 38.20 cm.
- Check: 100 ÷ 61.80 = 1.618 and 61.80 ÷ 38.20 = 1.618.
- A rectangle with sides 61.80 and 38.20 is a golden rectangle of area 2,360.68 cm².
Golden sections of common lengths
| Total | Longer part | Shorter part |
|---|---|---|
| 1 | 0.618 | 0.382 |
| 10 | 6.180 | 3.820 |
| 50 | 30.902 | 19.098 |
| 250 | 154.508 | 95.492 |
| 1,000 | 618.034 | 381.966 |
The link with Fibonacci numbers
In the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…, each number is the sum of the two before it. The ratio of consecutive numbers gets closer and closer to φ: 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, 55 ÷ 34 = 1.6176. This is why Fibonacci numbers give quick golden proportions with whole numbers, such as a 5 × 8 or 8 × 13 layout.
Using it in design
- Layouts: split a page width of 960 px into 593 px and 367 px columns.
- Typography: some designers scale font sizes by φ, for example 16 px body text with 26 px headings.
- Photography: the “phi grid” places lines at 38.2% and 61.8% of the frame instead of the thirds used by the rule of thirds.
The golden ratio is one pleasing proportion among many. Studies have not shown that people strongly prefer it over nearby ratios such as 3:2 or 16:10, so treat it as a useful tool rather than a law of beauty.