The golden ratio (φ ≈ 1.618) is a mathematical proportion that has fascinated mathematicians, artists, and architects for centuries. This calculator splits any length into the golden ratio, works backward from a known segment, and shows the matching golden-rectangle dimensions.

How the golden ratio works

Two quantities a and b (with a larger than b) are in the golden ratio when a/b equals (a+b)/a — the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part. Solving that equation algebraically gives φ = (1 + √5) / 2, an irrational number that starts 1.6180339887….

To split a total length into the golden ratio, divide it by φ to get the longer segment, then subtract that from the total to get the shorter segment. To go the other way — from a known segment to its golden-ratio partner — multiply by φ (to go from shorter to longer) or divide by φ (to go from longer to shorter).

Inputs and what they mean

The Split a Length calculation view accepts one number and a mode: whether that number is the total length, the longer segment, or the shorter segment. Any consistent unit works — inches, centimeters, pixels, dollars — because the golden ratio is a pure proportion with no units of its own.

The Golden Rectangle calculation view accepts a short side and multiplies it by φ to produce the long side and the resulting area. A golden rectangle has the property that removing a square from one end leaves a smaller rectangle that is itself a golden rectangle — a self-similar pattern sometimes associated with a logarithmic spiral.

Limits and edge cases

φ is irrational, so any decimal result is necessarily an approximation — this calculator uses the selected display precision, but the underlying math uses full floating-point precision. Entering zero or a negative number has no meaningful golden-ratio split and the calculator will prompt for a valid positive value instead.

The golden ratio's use in design and aesthetics is a long-standing convention rather than a physical law — treat the proportions here as a compositional starting point, not a rule that every layout or structure must follow.

Use finite values with magnitude at most 1e50. Expressions support pi, sqrt(), fractions and arithmetic. Angles are limited to 1e12 degrees. Units are mathematical; geometry alone does not establish physical safety. Read the linked values, named formulas, diagram coordinate table and any branch or parameter comparison.