Hong Life CalculatorsUnderstanding the golden ratio calculator — concept and principle
This calculator splits a length in the golden ratio, φ ≈ 1.618, or extends it. Enter the longer part, the shorter part or the whole, and it returns the other two lengths, the share of the shorter part and the matching golden rectangle. Use it for design layouts, photography crops, woodworking proportions and geometry problems.
How it is calculated
Golden ratio: (a + b) ÷ a = a ÷ b = φ = (1 + √5) ÷ 2 ≈ 1.6180339887
From the whole: a = whole ÷ φ, b = a ÷ φ
From the longer part: whole = a × φ, b = a ÷ φ
- The longer part is about 61.8 % of the whole, the shorter about 38.2 %.
- Golden rectangle: sides in ratio φ, here whole × (whole ÷ φ).
The principle
The golden ratio is the only proportion in which the whole relates to the larger part exactly as the larger part relates to the smaller. Writing that as an equation gives x² = x + 1, whose positive root is φ. Two neat consequences follow: φ² = φ + 1 and 1/φ = φ − 1 ≈ 0.618. Cutting a square from a golden rectangle leaves a smaller golden rectangle, and ratios of consecutive Fibonacci numbers (8/5, 13/8, 21/13, …) approach φ.
Worked example
Default inputs: the whole is known, length 100.
- Golden ratio φ = 1.618033989.
- Longer part a = 100 ÷ 1.618034 = 61.8034.
- Shorter part b = 61.8034 ÷ 1.618034 = 38.1966, which is 38.197 % of the whole.
- Check: a + b = 61.8034 + 38.1966 = 100.
- Golden rectangle for this length: 100 × 61.8034.
| Length (any unit) | Longer part a |
|---|---|
| 25 | 15.4508 |
| 50 | 30.9017 |
| 75 | 46.3525 |
| 100 | 61.8034 |
| 125 | 77.2542 |
| 150 | 92.7051 |
| 200 | 123.607 |
How to read the result
For a 100 cm shelf, a divider at about 61.8 cm from one end gives golden proportions. Many design rules of thumb use the simpler "rule of thirds" (33/67), which is close but not the same. Claims that famous buildings or paintings were designed on φ are often exaggerated; use it as one option for pleasant proportions, not as a law.
Common mistakes
- Dividing by 1.618 twice when only one step is needed.
- Using 0.618 and 1.618 the wrong way round.
- Rounding φ to 1.6, which gives 62.5 / 37.5 instead of 61.8 / 38.2.
- Treating the golden ratio as a rule of beauty that must be followed.
Frequently asked questions
What is the golden ratio?
φ = (1 + √5)/2 ≈ 1.618, the ratio in which whole ÷ larger part = larger part ÷ smaller part.
How do I divide 100 in the golden ratio?
About 61.8 and 38.2.
How is it related to the Fibonacci sequence?
The ratio of consecutive Fibonacci numbers approaches φ.
Sources: E. W. Weisstein, "Golden Ratio", Wolfram MathWorld