Golden Ratio Calculator
Phi, the golden ratio, is 1.6180339887… and one number unlocks five related jobs: splitting a line into golden sections, sizing a golden rectangle, finding the golden angle, drawing a golden spiral, and watching Fibonacci ratios converge to phi. Pick a mode below, enter the one value you know, and the calculator returns every related value with the steps shown.
Phi = (1 + sqrt(5)) / 2 = 1.6180339887498948… It is irrational, so every decimal you see is rounded.
Quick Answer
Split a line of 100 in the golden ratio and the longer segment is 61.803399 (61.80% of the whole) and the shorter is 38.196601 (38.20%). Check: 61.803399 ÷ 38.196601 = 1.618034 and 100 ÷ 61.803399 = 1.618034. The calculator is pre-filled with that split; switch modes for a golden rectangle (a longer side of 10 pairs with 6.18034), the golden angle (137.507764°), a drawn spiral, or the Fibonacci table.
Key takeaways
- One constant drives everything here: phi = (1 + √5) / 2 = 1.6180339887… Every mode is phi applied to a different object.
- The 61.8 / 38.2 split is scale-free. It holds for a 100-unit line, a 1-unit line (0.618034 and 0.381966), and a billion-unit line alike.
- Start from any one of the three segment lengths. Total, longer and shorter each determine the other two, because adjacent lengths differ by exactly one factor of phi.
- The golden angle and the golden spiral are the same mathematics in polar form: 137.507764° per step, radius multiplied by phi per quarter turn.
How to use this calculator
Pick a mode, enter the single value that mode needs, and click Calculate. Results stay hidden until then, and every mode prints its working steps under the answer.
- Golden section of a line: choose which length you know (total, longer or shorter), enter it, and get the other two lengths plus both phi checks.
- Golden rectangle: enter one side, say whether it is the longer or shorter side, and get the missing side, the area and the perimeter.
- Golden angle: the angle itself is a constant, so no input is required. Enter a circle radius only if you also want the two arc lengths it cuts.
- Golden spiral: enter a starting radius and a number of quarter turns (40 at most) and the tool computes the growth and draws the spiral on a canvas.
- Fibonacci convergence: no input at all. The tool tabulates the first 15 Fibonacci ratios so you can watch them settle onto phi.
Segment mode is the one most people need, and it is the same step-down, step-up pattern as a geometric sequence: multiply by phi to climb from shorter to longer to total, divide by phi to come back down. The common ratio calculator covers that repeated-multiplier idea for sequences in general.
The formulas behind the five modes
Each mode is one short formula set. Nothing in the tool uses an approximation of phi: the tool computes with phi = (1 + √5) / 2 at full double precision and rounds only for display, to six decimals in most outputs.
| Mode | Formula | You enter |
|---|---|---|
| Segment | Longer = Total ÷ phi; Shorter = Total − Longer (step up with × phi instead) | Any one of the three lengths |
| Rectangle | Longer ÷ Shorter = phi; Area = Longer × Shorter; Perimeter = 2 × (Longer + Shorter) | One side, and which side it is |
| Angle | Golden angle = 360° ÷ phi² = 137.507764° = 2.399963 rad; arcs = radius × angle in radians | Optional radius for arc lengths |
| Spiral | r = a × e^(bθ) with b = ln(phi) ÷ (π/2) = 0.306349; radius after q quarter turns = a × phi^q | Starting radius, quarter turns |
| Fibonacci | F(n+1) ÷ F(n) → phi as n grows; F(15) ÷ F(14) = 610 ÷ 377 = 1.61803714 | Nothing |
Two identities do most of the work and are worth knowing by hand: 1 ÷ phi = phi − 1 = 0.618034, and phi² = phi + 1 = 2.618034. The first is why dividing by phi and multiplying by 0.618034 give the same segment; the second is why the golden angle can be written as 360° × (2 − phi). Angle mode connects to ordinary circle work too: once you have an angle and a radius, arc length is just radius times radians, the same relationship the central angle calculator is built around.
Worked examples in every mode
Segment (the pre-filled run). Total 100: longer = 100 ÷ 1.618034 = 61.803399, shorter = 100 − 61.803399 = 38.196601. Starting from the shorter segment instead gives the same line back: 38.196601 × phi = 61.803399, and a shorter segment of 25 builds a line of 40.45085 + 25 = 65.45085.
Rectangle. Longer side 10: shorter = 10 ÷ phi = 6.18034, area = 10 × 6.18034 = 61.803399, perimeter = 2 × 16.18034 = 32.36068. Notice the area equals the longer segment of a 100 line; that is phi arithmetic reusing the same numbers, not a coincidence to read meaning into.
Angle. The golden angle is 137.507764° (2.399963 rad), its complement is 222.492236°, and it is 0.381966 of a full turn. With radius 10 the small arc is 10 × 2.399963 = 23.999632 and the large arc is 38.832221, summing to the circumference 62.831853; their ratio is phi, as the definition requires.
Spiral. Starting radius 1, eight quarter turns: the radius ends at 1 × phi^8 = 46.978714 after exactly two full turns, growing by the growth constant b = 0.306349 per radian. Doubling the start doubles every point on the curve: starting radius 2 over four quarter turns ends at 2 × phi^4 = 13.708204.
Fibonacci. The ratios oscillate around phi and close in: 13 ÷ 8 = 1.625, 21 ÷ 13 = 1.615385, 34 ÷ 21 = 1.619048, and by 377 ÷ 233 the ratio is 1.618026, within 0.000009 of phi. The table in Fibonacci mode shows all 15 steps, which is the honest way to present convergence: visible, term by term. Broader shape and proportion formulas live in the geometric formula calculator if you need the neighbouring geometry.
Phi in nature and design: what holds up and what does not
The defensible claim is narrow: phyllotaxis, the arrangement of leaves and seeds, genuinely follows the golden angle. Successive sunflower seeds emerge about 137.5° apart, which is why seed-head spiral counts land on consecutive Fibonacci numbers and why the packing wastes almost no space. That is a measured growth rule, not an aesthetic judgment.
| Popular claim | Verdict | Why |
|---|---|---|
| Sunflower and pinecone spirals follow Fibonacci counts | Holds up | Countable spirals in both directions, matching consecutive Fibonacci numbers, driven by the golden angle |
| The nautilus shell is a golden spiral | Does not hold up | Measured nautilus shells are logarithmic spirals, but they widen more slowly than phi per quarter turn |
| The Parthenon was designed on phi | Unsupported | Surveys find no consistent phi proportion; the fit depends on which points you choose to measure between |
| Phi proportions make designs objectively more beautiful | Unproven | Preference studies since Fechner have not found a reliable special preference for golden rectangles over nearby ratios |
None of this makes phi less useful as a design tool. A 61.8 / 38.2 split is a perfectly good, repeatable proportion system for layouts, type scales and photo crops; it just earns its place as a convention you chose, not a law of beauty you discovered. Use the numbers, skip the mysticism.
What competing calculators miss
The golden ratio keyword is served by tidy little two-box tools and a few broad ones. Each camp leaves something out:
- Omni Calculator: a polished segment and rectangle tool with a diagonal-based rectangle FAQ, but no golden angle, no spiral, and no Fibonacci table: three of this page’s five modes are absent.
- MiniWebTool: segment splitting with a small Fibonacci note. Correct as far as it goes, and it stops there: no rectangle area or perimeter, no angle, no drawing.
- OnlyCalculators: the closest rival, covering segments, rectangle, angle and Fibonacci in one page. It presents the worked example at four decimals and states some nature claims (nautilus, Parthenon) more warmly than the measurements support.
- GraphCalc: sections and rectangle in landscape and portrait orientations, a genuinely useful framing, but the content around the tool repeats the divine-proportion folklore (Parthenon, seashells) as fact.
- CalculatorUltra and the rectangle-only tools: one mode each. Fine if a rectangle is all you need, a dead end the moment the question is an angle or a spiral.
- The gap this page fills: all five phi jobs in one tool, full-precision phi with steps printed for every mode, a spiral that is actually drawn on a canvas rather than described, and a claims table that separates the measured phyllotaxis from the folklore.
Frequently asked questions
Phi = (1 + sqrt(5)) / 2, approximately 1.6180339887498948482… It is irrational, so its decimals never repeat; any value you see is rounded.
Because dividing the whole by phi gives the longer segment, and 1/phi = 0.61803… The shorter part is the remaining 38.20%, for any starting length.
Its sides are in the ratio phi. Cutting a square off one end leaves a smaller rectangle that is also golden, a self-similarity you can repeat forever.
The ratio of each Fibonacci number to the previous one converges to phi. Fibonacci mode in the calculator shows this convergence term by term.
About 137.5077 degrees: a full 360-degree turn divided in the golden ratio (360 / phi squared). Plants use it to arrange leaves so none exactly repeat, maximizing light exposure.
2.399963 radians, which is 137.507764 degrees. The tool shows both, plus the complementary 222.492236 degree angle and, when you enter a radius, the two arc lengths those angles cut.
Select the Golden spiral mode, enter a starting radius (for example 1) and the number of quarter turns (for example 8 for two full turns), then press Calculate. The tool shows the growth math and draws the spiral. The radius grows by exactly phi every quarter turn.
It appears genuinely in plant growth patterns like sunflower spirals. But some popular claims (nautilus shells, the Parthenon, faces) do not hold up to measurement. Treat each claim on its evidence.
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References
- Weisstein, Eric W., Wolfram MathWorld. Golden Ratio. Accessed October 21, 2026.
- Weisstein, Eric W., Wolfram MathWorld. Golden Rectangle. Accessed October 21, 2026.
- Weisstein, Eric W., Wolfram MathWorld. Golden Angle. Accessed October 21, 2026.
- Weisstein, Eric W., Wolfram MathWorld. Fibonacci Number. Accessed October 21, 2026.
- University of St Andrews, MacTutor History of Mathematics. The Golden Ratio. Accessed October 21, 2026.
- OEIS Foundation. A001622: Decimal expansion of the golden ratio phi. Accessed October 21, 2026.
- Encyclopaedia Britannica. Golden Ratio. Accessed October 21, 2026.