6.18034
16.18034
1.6180339887
1.6180339887
1.6180339887
0
2.6180339887
0.6180339887
137.5078
°
6.18034
16.18034
1.6180339887
1.6180339887
1.6180339887
0
2.6180339887
0.6180339887
137.5078
°
The Golden Ratio Calculator computes the golden proportion for any given segment length. The golden ratio $$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$$ is the unique positive number satisfying the proportion $$\frac{a}{b} = \frac{a + b}{a} = \varphi$$ — the ratio of the whole to the larger part equals the ratio of the larger part to the smaller.
The golden ratio has fascinated mathematicians since antiquity. Euclid defined it in Elements (c. 300 BCE) as "division in extreme and mean ratio." The German mathematician Martin Ohm coined the term "golden section" (goldener Schnitt) in 1835, and the symbol $$\varphi$$ (phi) honors the Greek sculptor Phidias, though this attribution is modern.
Algebraically, $$\varphi$$ is the positive root of $$x^2 - x - 1 = 0$$, giving $$\varphi = (1 + \sqrt{5})/2$$. This quadratic equation encodes the self-similar property: $$\varphi^2 = \varphi + 1$$, and equivalently $$1/\varphi = \varphi - 1$$. No other number has this remarkable reciprocal property — its reciprocal equals itself minus one.
The continued fraction representation $$\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\ddots}}} = [1; 1, 1, 1, \ldots]$$ consists entirely of ones, making $$\varphi$$ the "most slowly convergent" continued fraction and, in a precise sense, the most irrational number. This property explains its appearance in phyllotaxis — the arrangement of leaves and seeds in plants. When successive elements are placed at the golden angle $$360°/\varphi^2 \approx 137.508°$$, they achieve the most uniform distribution, avoiding the clumping that rational angle fractions would cause.
The golden ratio is intimately connected to the Fibonacci sequence: $$\lim_{n \to \infty} F_{n+1}/F_n = \varphi$$. In geometry, a golden rectangle (sides in ratio $$\varphi : 1$$) can be subdivided into a square and a smaller golden rectangle, creating a self-similar tiling that spirals inward as the logarithmic spiral. The regular pentagon and pentagram are rich with golden ratios — the diagonal-to-side ratio of a regular pentagon is exactly $$\varphi$$.
In art and architecture, proportions close to the golden ratio appear in the Parthenon, Le Corbusier's Modulor system, and compositions by Leonardo da Vinci. While some claims about its prevalence are exaggerated, the mathematical properties of $$\varphi$$ — irrationality, self-similarity, optimal spacing — make it genuinely special among all real numbers.
The calculator divides a given length $$a$$ in the golden ratio and verifies the defining proportion.
Step 1: Compute φ.
$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$$
Step 2: Find the shorter segment.
$$b = \frac{a}{\varphi}$$
Step 3: Verify the golden proportion. Two ratios should be equal:
$$\frac{a}{b} = \varphi \quad \text{and} \quad \frac{a + b}{a} = \varphi$$
Step 4: Compute derived constants.
$$\varphi^2 = \varphi + 1 \approx 2.6180339887$$
$$\frac{1}{\varphi} = \varphi - 1 \approx 0.6180339887$$
Step 5: Golden angle.
$$\theta = \frac{360°}{\varphi^2} \approx 137.508°$$
This is the angle that produces optimal spacing in spiral phyllotaxis.
φ is the golden ratio constant, approximately 1.6180339887. It is irrational and algebraic (degree 2 over the rationals).
Shorter Segment b is the smaller piece when the input $$a$$ is divided in the golden ratio. Together, $$a$$ and $$b$$ form a golden section.
Total a + b is the full length. The ratio $$(a+b)/a$$ should equal $$a/b = \varphi$$, confirming the defining property.
Verification checks that both ratios agree. Since $$b$$ is computed as $$a/\varphi$$, this is always satisfied to floating-point precision.
φ² = φ + 1 demonstrates the fundamental quadratic property. Numerically, $$2.618... = 1.618... + 1$$.
1/φ = φ − 1 shows the unique reciprocal property. The decimal digits of $$\varphi$$ and $$1/\varphi$$ match after the decimal point: 0.6180339887...
Golden Angle ≈ 137.508° is the smaller of the two angles created by dividing a full circle in the golden ratio. It governs the spiral patterns in sunflower heads and pinecones.
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A segment of length 10 divided in the golden ratio gives b ≈ 6.18. Both ratios 10/6.18 and 16.18/10 equal φ ≈ 1.618. The golden angle is approximately 137.5°.
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A golden rectangle with longer side 100 has shorter side ≈ 61.8. The rectangle can be tiled by removing a 61.8 × 61.8 square, leaving a smaller golden rectangle of dimensions 61.8 × 38.2, and so on infinitely.
The continued fraction of $$\varphi = [1; 1, 1, 1, \ldots]$$ has the smallest possible partial quotients (all 1s), meaning its rational approximations converge as slowly as possible. By Hurwitz's theorem, for any irrational $$\alpha$$, there are infinitely many rationals $$p/q$$ with $$|\alpha - p/q| < 1/(\sqrt{5}q^2)$$, and the constant $$\sqrt{5}$$ is best possible precisely because of $$\varphi$$. Numbers that are hardest to approximate by rationals are in this sense "most irrational."
In a regular pentagon with side length 1, the diagonal has length $$\varphi$$. Each diagonal is divided by the others in the golden ratio. The pentagram (five-pointed star) formed by the diagonals creates a smaller regular pentagon at its center, with side length $$1/\varphi^2$$. This self-similar structure repeats infinitely, and every length ratio in the figure is a power of $$\varphi$$.
A golden spiral is a logarithmic spiral whose growth factor is $$\varphi$$ per quarter turn. It is constructed by drawing circular arcs through the corners of nested golden rectangles. The spiral closely approximates the Fibonacci spiral (arcs through Fibonacci-number squares) but is a true equiangular spiral with a constant angle of approximately 72.97° with respect to radial lines.
The golden ratio appears in some works (Le Corbusier intentionally used it in his Modulor system), but many claimed sightings in the Parthenon, pyramids, and paintings rely on selective measurement and confirmation bias. Mathematician Keith Devlin has cautioned against overstating its aesthetic significance. However, $$\varphi$$'s genuine mathematical properties — self-similarity, optimal irrationality, connection to Fibonacci numbers — ensure its importance independently of aesthetic claims.
The ratio $$F_{n+1}/F_n$$ converges to $$\varphi$$ as $$n \to \infty$$. More precisely, Binet's formula gives $$F_n = (\varphi^n - \psi^n)/\sqrt{5}$$ where $$\psi = -1/\varphi$$. The Fibonacci sequence can be viewed as the integer sequence that best approximates the geometric sequence $$\varphi^n/\sqrt{5}$$, and the golden ratio is the eigenvalue of the Fibonacci matrix $$\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$.
The golden angle $$\approx 137.508°$$ is used in phyllotaxis — the study of leaf and seed arrangements in plants. When successive organs are placed at this angle around a central axis, they achieve the most uniform distribution with minimal gaps, because $$\varphi$$ is the hardest number to approximate by rationals (no nearby rational rotation would create aligned rows). This is why sunflower seed counts in spirals typically follow consecutive Fibonacci numbers.
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