What this calculator does
The golden ratio is the proportion where the whole relates to the larger part exactly as the larger part relates to the smaller. That self-similarity is what makes it unusual: it is the only positive ratio with that property.
The arithmetic is neat. Splitting a total of 10 gives 6.180 and 3.820, and dividing one by the other returns 1.618 exactly, which is also what you get dividing the total by the larger part.
The formula
Phi is one plus the square root of five, all over two. Depending on which part is known, the other is found by multiplying or dividing by phi, or by splitting the total in that proportion.
| Term | Meaning |
|---|---|
| Phi (φ) | Approximately 1.618, the golden ratio. |
| Self-similarity | The whole to the larger part equals the larger to the smaller. |
| Golden rectangle | A rectangle whose sides are in this proportion. |
The inputs explained
| Field | What to enter |
|---|---|
| Length | The length you know. |
| This length is | Whether that length is the shorter part, the longer part, or the total to be divided. |
When to use it
Laying out a design
The proportion is widely used in typography and layout, though its aesthetic claims are disputed.
Dividing a length
Splitting a total in golden proportion takes one calculation.
Understanding the ratio
Seeing the same 1.618 appear in both comparisons demonstrates the self-similarity.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does each mode divide the length?
The same number treated three different ways.
| The length of 10 is | Longer part | Shorter part |
|---|---|---|
| The shorter part | 16.180 | 10.000 |
| The longer part | 10.000 | 6.180 |
| The total | 6.180 | 3.820 |
Questions
Where does the value come from?
From solving the defining proportion, which reduces to a quadratic. Its positive solution is one plus the square root of five over two, an irrational number approximately equal to 1.618.
What is special about phi?
Several things. Its reciprocal is itself minus one, and its square is itself plus one. No other number has that pair of properties, and both follow directly from the defining quadratic.
Does it really appear throughout nature and art?
Less than commonly claimed. It genuinely appears in phyllotaxis, the spiral arrangement of plant leaves and seeds, for sound mathematical reasons. Many claims about the Parthenon and Renaissance painting do not survive measurement.
How does it relate to Fibonacci numbers?
The ratio of consecutive Fibonacci numbers converges to phi. That convergence is fast: by the tenth term the ratio is already correct to three decimal places.
For the sequence that converges to it, see the Fibonacci calculator. For sequence arithmetic generally, see the geometric sequence calculator.