Golden Ratio: worked example
| The length you know is the | Longer part |
|---|---|
| Length | 10 |
| Longer part | 10 |
| Shorter part | 6.2 |
| Whole length | 16.2 |
Common error: Splitting a length by 60 and 40 percent and calling it golden. The true split is about 61.8 and 38.2 percent, close but not the same.
Golden Ratio: formula
The ratio is irrational, so a finite decimal display is rounded. In a physical layout, material dimensions, tolerances and function remain independent requirements. Rounding a proposed part to a whole measurement changes the resulting proportion.
Choose which length you know, the longer part, the shorter part or the whole, and enter it. The golden ratio, about 1.618034, is (1 plus the square root of 5) over 2. Two parts are in golden proportion when the whole is to the longer part as the longer is to the shorter.
The tool then finds the other two lengths. Use consistent units.
Golden Ratio: changing Length
| Length | Longer part | Shorter part | Whole length |
|---|---|---|---|
| 8 | 8 | 4.9 | 12.9 |
| 9 | 9 | 5.6 | 14.6 |
| 10 (example) | 10 | 6.2 | 16.2 |
| 11 | 11 | 6.8 | 17.8 |
| 12 | 12 | 7.4 | 19.4 |
Golden Ratio: compare input ranges
| Input | Tested from | To | Longer part: low input | Longer part: high input | Swing |
|---|---|---|---|---|---|
| The length you know is the | Longer part | Whole length | 10 | 6.2 | 3.8 (100%) |
| Length | 9 | 11 | 9 | 11 | 2 (20%) |
Golden Ratio: individual input comparisons
| Input | Value | Longer part | Shorter part | Whole length |
|---|---|---|---|---|
| The length you know is the | Longer part | 10 | 6.2 | 16.2 |
| Shorter part | 16.2 | 10 | 26.2 | |
| Whole length | 6.2 | 3.8 | 10 | |
| Length | 8 | 8 | 4.9 | 12.9 |
| 9 | 9 | 5.6 | 14.6 | |
| 10 | 10 | 6.2 | 16.2 | |
| 11 | 11 | 6.8 | 17.8 | |
| 12 | 12 | 7.4 | 19.4 |
Golden Ratio: which assumptions move the result?
Golden Ratio: values and definitions
| Quantity and definition | Value used |
|---|---|
| The length you know is the — Choose which of the three lengths you are entering. | Longer part |
| Length | 10 |
| Longer part | 10 |
| Shorter part | 6.2 |
| Whole length | 16.2 |
Geometry calculators: Midpoint Calculator, Rectangle Area & Perimeter Calculator, Circle Area & Circumference Calculator, Triangle Area (Heron's Formula) Calculator, Sphere Volume & Surface Area Calculator, Cylinder Volume & Surface Area Calculator.
The golden ratio is the positive solution of φ² − φ − 1 = 0, approximately 1.618. Its square equals φ + 1 and its reciprocal equals φ − 1. Those identities keep the proportions consistent when a length is divided repeatedly; they do not establish that a particular design is objectively preferable.
A whole length, a longer segment and a shorter segment describe different quantities; the same entered number represents a different starting length in each case.
See how consecutive Fibonacci numbers approach it on the Fibonacci calculator, and general ratios on the ratio simplifier and proportion calculators. The rectangle area and regular polygon calculators help lay out shapes, and the TV size page relates to aspect ratios.
What is the golden ratio?
About 1.618, the ratio at which a line divided into two parts has the whole to the longer part in the same proportion as the longer part to the shorter.
How is the golden ratio related to the Fibonacci sequence?
The ratio of consecutive Fibonacci numbers gets closer to the golden ratio as the numbers grow.
Do I need to use it in design?
No. It is one option among many, and good design depends on function and balance as well as proportion.