Enter any one value — A (longer segment), B (shorter segment), or A+B (total length) — and the Golden Ratio Calculator solves for all three. You'll see how your segment splits according to phi (φ ≈ 1.618), with a visual breakdown of the proportions.
Golden Ratio (φ)
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Longer Segment (A)
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Shorter Segment (B)
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Total Length (A+B)
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(A+B) / A
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A / B
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The golden ratio (φ) is an irrational number approximately equal to 1.6180339887. It arises when a line segment is divided into two parts such that the ratio of the whole segment to the longer part equals the ratio of the longer part to the shorter part. It's often denoted by the Greek letter phi (φ) and is found extensively in mathematics, art, and nature.
Divide the longer segment (A) by the shorter segment (B). If the result is approximately 1.618, the two segments are in the golden ratio. You can also check by dividing the total length (A+B) by A — this should also equal approximately 1.618.
Simply enter any one of the three values — the total length (A+B), the longer segment (A), or the shorter segment (B). The calculator will automatically solve for the other two values using phi (φ ≈ 1.618). You can also adjust the number of decimal places shown.
The golden ratio appears across disciplines — from the proportions of ancient Greek architecture and Renaissance paintings to spiral patterns in seashells and galaxies. Designers and artists have long used it as a guideline for aesthetically pleasing proportions, and it has deep connections to the Fibonacci sequence in mathematics.
The golden ratio appears in the spiral arrangement of sunflower seeds, the branching of trees, the curve of nautilus shells, and even the proportions of the human body. It's closely related to the Fibonacci sequence, where consecutive Fibonacci numbers (e.g. 8, 13, 21) produce ratios that converge toward φ.
A golden rectangle is a rectangle whose side lengths are in the golden ratio — that is, the ratio of the longer side to the shorter side equals φ ≈ 1.618. If you remove a square from a golden rectangle, the remaining piece is itself a golden rectangle, and this process can repeat infinitely, producing the golden spiral.
The golden ratio is defined as φ = (1 + √5) / 2, which equals approximately 1.6180339887499. It satisfies the unique property that φ² = φ + 1, and 1/φ = φ − 1 ≈ 0.618.
For a golden rectangle with diagonal d = 1, the sides are A = √(φ² / (1 + φ²)) and B = √(1 / (1 + φ²)). Numerically, the longer side is approximately 0.8507 and the shorter side is approximately 0.5257.