Golden Rectangle Calculator

A golden rectangle has sides in the ratio 1 : φ where φ = (1+√5)/2 ≈ 1.61803398…

The golden ratio has a unique property: if you cut a square from a golden rectangle, the remaining piece is another golden rectangle. You can repeat this forever, and the nested squares carry a quarter circle each that join into a smooth logarithmic spiral.

A caveat on that spiral, since it is the most over-claimed picture in mathematics. The golden spiral grows by a factor of φ every quarter turn, which is steep. Nautilus shells, galaxy arms and hurricane bands are all logarithmic spirals, but with their own growth rates: a nautilus expands by roughly 1.3 per turn, nothing like φ. Overlaying a golden spiral on a nautilus photograph and declaring a match requires generous scaling. The spiral below is real mathematics; the nature photographs usually are not.

The exact value

φ = (1+√5)/2

This is the positive root of x² - x - 1 = 0. It satisfies φ = 1 + 1/φ, which means you can write it as an infinite continued fraction: φ = 1 + 1/(1 + 1/(1 + 1/…)). It is the hardest number to approximate with simple fractions, which makes it the most “irrational” of all irrationals.

Fibonacci and the golden ratio

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21…) converges to φ: each term divided by the previous term approaches 1.61803… as the sequence grows. This is why Fibonacci numbers appear in phyllotaxis (leaf arrangement), pine cones, and sunflower seed spirals.

Does it appear in art?

The claim that the golden ratio appears in the Parthenon, Leonardo’s paintings, and other classical works is largely a myth propagated in the 20th century. Most such claims involve cherry-picking measurements until they fit. The ratio does appear in architecture and design deliberately when artists consciously apply it.

Diagonal

The diagonal of a golden rectangle with shorter side w is √(w² + (φw)²) = w√(1+φ²) = w√(φ+2) ≈ 1.902w.

Enter either dimension to get the full rectangle.