What this calculator does
A golden rectangle is one whose sides stand in the golden ratio, φ, approximately 1.618034. Give it a short side and the long side follows by multiplying by φ. There is nothing to choose or adjust, because the shape is fixed and only its size varies.
Its defining property is self-similarity. Cut a square off one end of a golden rectangle and what remains is another golden rectangle, smaller but exactly the same shape, and the cut can be repeated forever. Those nested squares are what the golden spiral is usually drawn through. Claims that the proportion runs through classical art and architecture are mostly retrofitted and rarely survive careful measurement, but the geometry itself is real enough.
The formula
The short side entered is multiplied by φ, which is (1 + √5) ÷ 2, to give the long side. The diagonal is then the ordinary Pythagorean result, the square root of the two sides squared and added together. The area is simply the two sides multiplied. The ratio is shown as a fourth figure so it can be seen holding steady no matter what size is entered.
| Term | Meaning |
|---|---|
| φ (phi) | The golden ratio, (1 + √5) ÷ 2, approximately 1.618034. It is irrational, so it has no exact decimal form. |
| Golden rectangle | A rectangle whose long side divided by its short side equals φ. |
| Self-similarity | The property that removing a square from a golden rectangle leaves a smaller rectangle of the same proportions. |
| Diagonal | The corner-to-corner distance, found from the two sides by Pythagoras. |
The inputs explained
| Field | What to enter |
|---|---|
| Short side (width) | The short side of the rectangle, in whatever unit you are working in. The long side, diagonal and area all come back in the same unit or its square. |
When to use it
Laying out a design to fixed proportions
A page, a canvas or a frame set to the golden proportion needs only one dimension chosen, with the other following from it. Whether the proportion looks better than any nearby one is a matter of taste, but it is at least a consistent choice.
Drawing the golden spiral
The spiral is built by repeatedly cutting a square from the rectangle and drawing a quarter circle inside each square. Having the side lengths at each step makes the construction straightforward to set out accurately.
Working through a geometry exercise
Golden rectangle problems usually ask for the remaining side, the diagonal or the area from one given dimension, and all three come back together here.
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 do the sides, diagonal and area scale with the short side?
Five golden rectangles, differing only in scale.
| Short side | Long side | Diagonal | Area | Ratio (long ÷ short) |
|---|---|---|---|---|
| 1 | 1.618 | 1.902 | 1.618 | 1.6180 |
| 5 | 8.090 | 9.511 | 40.451 | 1.6180 |
| 10 | 16.180 | 19.021 | 161.803 | 1.6180 |
| 16 | 25.889 | 30.434 | 414.217 | 1.6180 |
| 100 | 161.803 | 190.211 | 16,180.34 | 1.6180 |
Questions
What is the golden ratio exactly?
It is (1 + √5) ÷ 2, the positive solution of x² = x + 1. Because √5 is irrational, φ is too, so 1.618034 is a rounding rather than the value itself. It is the only positive number whose square is itself plus one, and equally the only one whose reciprocal is itself minus one.
Why is a golden rectangle called self-similar?
Because cutting a square off the end leaves a rectangle with the same proportions as the original. That happens only at the golden ratio, and it is the property the whole shape is defined by rather than a coincidence about it.
Is A4 paper a golden rectangle?
No. A-series paper uses a ratio of √2, about 1.414, chosen so that folding a sheet in half gives the next size down with the same proportions. That is a different self-similarity from the golden one, and it produces a noticeably narrower rectangle.
Does the golden rectangle really appear throughout art and nature?
Far less than it is claimed to. Some Fibonacci counts in plants are genuine, since they come from efficient packing. Most assertions about paintings, temples and shells rely on choosing the measurement points after the fact, and rarely hold up when the same measurements are taken independently.
For the ratio itself and its relationship to the Fibonacci numbers, see the golden ratio calculator. For the sequence whose successive ratios converge on φ, see the Fibonacci sequence calculator.