Maths / Design
Golden Ratio Calculator (φ ≈ 1.618)
Split a length in the golden ratio, or find the other part and the whole from one part, with the matching golden rectangle and a drawing of its spiral.
Golden Ratio Calculator (φ ≈ 1.618): The golden ratio φ = (1 + √5) ÷ 2 ≈ 1.6180 is the ratio where the whole is to the longer part as the longer part is to the shorter. A length of 100 divides into 61.80 and 38.20; the longer part is the whole ÷ φ, and the shorter the longer ÷ φ. A golden rectangle has sides in the same ratio. Runs 100% locally in your browser with zero server file uploads.
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Two lengths are in the golden ratio when the whole is to the longer part as the longer part is to the shorter: (a + b) ÷ a = a ÷ b = φ = (1 + √5) ÷ 2 ≈ 1.6180. A golden rectangle cut into a square leaves a smaller golden rectangle, and so on, which traces the spiral in the drawing. Ratios of consecutive Fibonacci numbers, 8 ÷ 5, 13 ÷ 8, 21 ÷ 13, approach φ.
In design
A 1,200-pixel layout split in the golden ratio gives columns of about 742 and 458 pixels. Type scales sometimes multiply each heading size by 1.618, which grows quickly: 16, 26, 42 px.
For other proportions such as 16:9 or 4:3, use the aspect ratio calculator.
Approximations
Simple fractions close to φ include 8/5 = 1.6, 13/8 = 1.625, and 21/13 ≈ 1.615. The A paper sizes use a different ratio, √2 ≈ 1.414, chosen so that halving a sheet keeps its shape.
To work with exact fractions, use the fraction calculator.
How to use it
- Choose whether you know the whole length, the longer part, or the shorter part.
- Enter the length.
- Read the other two lengths and the golden rectangle's sides.
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Related tools
Frequently asked questions
Why is φ special?
It is the only positive number whose reciprocal is itself minus 1 (1/φ = φ − 1 ≈ 0.618) and whose square is itself plus 1 (φ² = φ + 1 ≈ 2.618).
How is it related to Fibonacci numbers?
Ratios of consecutive Fibonacci numbers, 5 ÷ 3, 8 ÷ 5, 13 ÷ 8, 21 ÷ 13, get ever closer to φ.
Is it really in famous buildings and paintings?
Many claims are exaggerated: measurements can usually be chosen to fit. It is a useful proportion in design, not a hidden law.
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