Golden ratio calculator
Enter a length, get its golden-ratio parts.
What is the golden ratio?
The number φ = (1 + √5) ÷ 2 ≈ 1.6180339887. Two lengths are in the golden ratio when the whole is to the longer part as the longer part is to the shorter.
How do I split a length in the golden ratio?
Divide it by φ to get the longer part; the rest is the shorter part. A 10-inch line splits into 6.180 in and 3.820 in, and 10 ÷ 6.180 = 6.180 ÷ 3.820 = 1.618.
What if I know the shorter side?
Multiply by φ to get the longer side: a rectangle 10 units tall is 16.18 units wide in golden proportion.
What special property does φ have?
φ² = φ + 1 and 1/φ = φ − 1 ≈ 0.618, so its square and its reciprocal have the same decimals as φ itself.
How is it related to Fibonacci numbers?
The ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21 …) gets closer and closer to φ: 21 ÷ 13 ≈ 1.615, 34 ÷ 21 ≈ 1.619.
Is the golden ratio a fraction?
No. φ = (1 + √5) ÷ 2 and √5 is irrational, so the decimals never end or repeat. MathWorld calls φ algebraic of degree 2 and notes its continued fraction [1; 1, 1, 1, …] makes it unusually hard to approximate by fractions. Fibonacci fractions come close: 89 ÷ 55 = 1.61818 is off by only 0.00015.
Is the golden ratio really the most pleasing proportion?
Not as a rule. MathWorld notes that claims about φ in art, architecture, sculpture and anatomy tend to be greatly exaggerated; George Markowsky’s “Misconceptions about the Golden Ratio” (The College Mathematics Journal, 1992) examines several popular ones. Treat 1 : 1.618 as one proportion to try, not a law of beauty.
The formulas and what they assume
A segment is divided in the golden ratio when the whole is to the longer part as the longer part is to the shorter (Wolfram MathWorld, “Golden Ratio”). Calling the shorter part 1 and the longer x gives (x + 1) ÷ x = x, so x² − x − 1 = 0 and the positive root is φ = (1 + √5) ÷ 2 = 1.6180339887. For a whole length a the calculator returns the longer part a ÷ φ and the shorter part a ÷ φ², which add back to a because φ² = φ + 1. A golden rectangle has sides in the ratio 1 : φ. The answers are in the unit you enter, and the tool uses the full value of φ, not 1.618.
Worked example: a 60-inch wall
The longer part is 60 ÷ φ = 37.082 in and the shorter is 60 ÷ φ² = 22.918 in, which together make 60. Both ratios check: 60 ÷ 37.082 and 37.082 ÷ 22.918 are 1.618. A golden rectangle 8 inches tall is 8 × φ = 12.944 in long; cut off an 8-inch square and what remains, 4.944 by 8, is again a golden rectangle, as MathWorld describes.
Common mistakes
Treating 1.618 as exact: MathWorld lists calling φ “exactly 1.618” as an error. The gap is small but real, since 1000 ÷ 1.62 is 617.3 and 1000 ÷ φ is 618.0. Calling a ÷ φ the shorter part: it is the longer part, and the shorter one needs φ². And expecting a layout to look better only because it follows 1 : 1.618, as the last question explains.
Sources: Wolfram MathWorld, “Golden Ratio” (definition, φ² = φ + 1, Fibonacci limit, golden rectangle, remarks on exaggerated claims); G. Markowsky, “Misconceptions about the Golden Ratio”, The College Mathematics Journal 23 (1992). Related tools: Fibonacci calculator, quadratic formula calculator, ratio calculator and aspect ratio calculator. How we check our calculators.