Definition and calculation
Divide a segment of length 1 into a longer part x and a shorter part 1−x so that 1 : x = x : (1−x); the ratio of the whole to the longer part is then φ = (1+√5)/2 ≈ 1.618, the golden ratio. Around 300 BC Euclid's Elements called it the "extreme and mean ratio" and showed how to construct it; the name "golden section" spread in 19th-century Germany. φ satisfies φ² = φ + 1, so squaring it adds one (2.618) and its reciprocal subtracts one (0.618). Written as a continued fraction it is [1; 1, 1, 1, …], which makes it the irrational number hardest to approximate by fractions.
The pentagon and the Fibonacci sequence
In a regular pentagon the ratio of a diagonal to a side is exactly φ, and the diagonals cut one another in the golden ratio, which is why the pentagram became a symbol of the Pythagoreans. Cut a square from a golden rectangle (sides φ:1) and another golden rectangle remains; repeating this and joining quarter-circles draws the golden spiral. Ratios of neighbouring terms of the Fibonacci sequence (8/5, 13/8, 21/13…) converge on φ, and the "golden angle" of about 137.5 degrees at which many plants place successive leaves is also derived from φ.
Myths and real uses
Popular claims that the Parthenon, the pyramids, the Mona Lisa or the human body were designed on the golden ratio usually rely on choosing convenient measuring points, and historians and mathematicians have criticised them as poorly supported. Luca Pacioli published De divina proportione in 1509 with illustrations by Leonardo da Vinci, but there is no evidence Leonardo deliberately built the ratio into his paintings. Since Gustav Fechner's experiments in the 1870s, studies of whether people prefer golden rectangles have produced inconsistent results. There are genuine deliberate uses, such as Le Corbusier's Modulor system of architectural proportions, and today the ratio is one of several guides for logos, photo composition and web layouts.
FAQ
- How do you calculate the golden ratio?
- It is the positive solution of the quadratic equation x² = x + 1. The quadratic formula gives (1+√5)/2 ≈ 1.618. Dividing neighbouring Fibonacci numbers also gives close approximations.
- Are credit cards or A4 paper in the golden ratio?
- A credit card (about 1.586) is close but not exact. A4 paper uses √2 ≈ 1.414 so that cutting it in half keeps the same shape, which is a different ratio.
Sources reviewed: 2026-10-26
References
- Heath, T. L. (1956). The Thirteen Books of Euclid's Elements (2nd ed., Vol. 2). Dover.
- Livio, M. (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. Broadway Books.
- Markowsky, G. (1992). Misconceptions about the golden ratio. The College Mathematics Journal, 23(1), 2–19.
- Herz-Fischler, R. (1998). A Mathematical History of the Golden Number. Dover.