Enter any one value — Side A (longer), Side B (shorter), or Sum (A+B) — and the Golden Section Calculator fills in the remaining proportions using the golden ratio (φ = 1.618). You'll see all three segment lengths, the exact ratio, and a visual breakdown of how the values relate to each other.
Golden Ratio (A/B)
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Side A (Longer)
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Side B (Shorter)
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Sum A + B
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(A+B)/A Ratio
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The golden ratio, also known as the golden section or golden mean, is a special mathematical proportion approximately equal to 1.6180339887 (φ). It occurs when a 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 appears throughout art, architecture, nature, and design.
Simply enter any ONE of the three values — Side A (longer), Side B (shorter), or the total Sum (A+B) — and leave the others blank. The calculator will automatically compute the remaining two values using the golden ratio formula φ = (1 + √5) / 2 ≈ 1.618.
Divide the longer segment by the shorter segment. If the result is approximately 1.618, the two segments are in the golden ratio. Equivalently, dividing the total length (A+B) by A should also give approximately 1.618. Enter both values into this calculator to verify instantly.
The core formulas are: Side A = Side B × 1.618, Side B = Side A ÷ 1.618, Side A = Sum(A+B) × 0.618, Side B = Sum(A+B) × 0.382, and Sum(A+B) = Side A + Side B. These all derive from φ = (1 + √5) / 2.
The golden ratio is thought to create naturally harmonious proportions that the human eye finds balanced and beautiful. It has been intentionally used in famous works of art and architecture, including the Parthenon, Leonardo da Vinci's paintings, and many modern designs. Some researchers also link it to the sense of balance found in natural forms.
The golden ratio and the related Fibonacci sequence appear in sunflower seed spirals, nautilus shells, pine cone arrangements, the branching of trees, and the proportions of the human body. These patterns arise because φ-based growth allows for the most efficient packing and structural strength.
A golden rectangle is one whose side lengths are in the golden ratio — the longer side divided by the shorter side equals approximately 1.618. When you remove a square from a golden rectangle, the remaining piece is itself another golden rectangle, a property that repeats infinitely and produces the famous golden spiral.
The exact value of phi (φ) is (1 + √5) / 2, which equals 1.6180339887499... — an irrational number that never ends or repeats. For most practical design and calculation purposes, 1.618 is a sufficient approximation.