Golden Section Calculator
Divide a line so the whole-to-longer ratio equals the longer-to-shorter ratio. MiniWebtool’s Golden Section Calculator finds the missing lengths from the total or either segment and explains the golden-ratio relationship with a diagram.
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Enter the total length a + b, the long part a or the short part b below to get the other two, with the steps and a diagram.
Golden Section Result
Visual Representation
Step-by-Step Calculation
a = 6 / 1.6180339887 = 3.7082039325
b = 3.7082039325 / 1.6180339887 = 2.2917960675
Check: a / b = 3.7082039325 / 2.2917960675 = 1.6180339887
Both ratios equal φ ≈ 1.618, confirming the golden section.
About Golden Section Calculator
Welcome to the Golden Section Calculator, an elegant tool for exploring the divine proportion. Calculate the missing segments of a line divided according to the golden ratio (φ ≈ 1.6180339887), visualize the proportions with interactive diagrams, and discover the mathematical beauty that appears throughout nature, art, and architecture.
At a Glance
The Golden Section Calculator divides a line in the golden ratio. Enter the total length, the longer segment or the shorter segment and it returns the other two values.
| Feature | What you get | Good for |
|---|---|---|
| One known value | Enter any one of a+b, a or b and leave the other fields empty to get all three values. | Finding a missing length from a single measurement |
| Golden ratio formulas | Uses (a+b)/a = a/b = phi, about 1.6180339887, for example a = b x phi and b = a x 0.618. | Applying the section to a layout or drawing |
| Worked example | With a = 10, the calculator gives b about 6.18 and a+b about 16.18. | Sanity-checking results |
| SVG visualization | A diagram with a golden spiral shows the proportions of the divided line. | Seeing how the two segments compare |
| Step-by-step derivation | Shows the mathematical derivation of the results. | Following the working for coursework |
What is the Golden Section?
The golden section (also called the golden cut or divine proportion) is a way of dividing a line segment so that the ratio of the whole line to the longer segment equals the ratio of the longer segment to the shorter segment. This special ratio is known as the golden ratio, denoted by the Greek letter phi (φ).
When a line segment is divided at the golden ratio, the relationship creates a unique mathematical harmony. The longer segment a is to the shorter segment b as the whole line (a + b) is to the longer segment a.
Mathematical Properties of φ
The golden ratio possesses remarkable mathematical properties that have fascinated mathematicians for millennia:
How to Use This Calculator
- Choose your known value: Decide whether you know the total length (a+b), the longer segment (a), or the shorter segment (b).
- Enter the value: Input your known value into the corresponding field. Leave the other fields empty.
- Calculate: Click the Calculate button to find all three golden section values.
- View results: See the calculated values and the interactive visualization showing the golden proportions.
The Golden Ratio in Nature
The golden ratio appears with remarkable frequency in the natural world, earning it the name "divine proportion":
Fibonacci Sequence and φ
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, ...) has a profound connection to the golden ratio. As the sequence progresses, the ratio of consecutive terms approaches φ:
| Fibonacci Numbers | Ratio | Approaches φ |
|---|---|---|
| 3 / 2 | 1.500 | → φ |
| 5 / 3 | 1.667 | → φ |
| 8 / 5 | 1.600 | → φ |
| 13 / 8 | 1.625 | → φ |
| 21 / 13 | 1.615 | → φ |
| 34 / 21 | 1.619 | → φ |
| 55 / 34 | 1.618 | ≈ φ |
Golden Ratio Formulas
Finding a from b
Finding b from a
Finding a+b from a
Frequently Asked Questions
What is the golden ratio?
The golden ratio (φ ≈ 1.6180339887) is an irrational number that appears when a line is divided into two parts such that the ratio of the whole line to the longer part equals the ratio of the longer part to the shorter part. It is found throughout nature, art, and architecture.
How do I calculate golden section values?
Enter any one value (a+b, a, or b) into the calculator. The tool will automatically compute the other two values using the golden ratio formula: (a+b)/a = a/b = φ ≈ 1.618. For example, if a = 10, then b = a/φ ≈ 6.18 and a+b ≈ 16.18.
Where is the golden ratio found in nature?
The golden ratio appears in sunflower seed spirals, nautilus shells, hurricane formations, galaxies, flower petals (often Fibonacci numbers which converge to φ), pinecones, and the proportions of many animals and plants. It is also used extensively in art, architecture, and design.
What is the relationship between Fibonacci and the golden ratio?
The ratio of consecutive Fibonacci numbers approaches the golden ratio as the numbers get larger. For example: 5/3 = 1.667, 8/5 = 1.600, 13/8 = 1.625, 21/13 = 1.615, and so on, converging toward φ ≈ 1.6180339887.
Additional Resources
Reference this content, page, or tool as:
"Golden Section Calculator" at https://MiniWebtool.com/golden-section-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: Jan 25, 2026
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