Quick answer
The golden ratio φ ≈ 1.6180339887… A line split in golden ratio: longer part ÷ shorter part = total ÷ longer part = φ. Given the longer side a: shorter side = a ÷ 1.618, total = a × 1.618.
How to use this calculator
Select whether you know the longer side, the shorter side, or the total length. Enter the value. The calculator returns all three dimensions (longer, shorter, total) and the actual ratio a/b to confirm it equals φ.
What is the golden ratio?
The golden ratio φ (phi) is an irrational number approximately equal to 1.618. It is defined by the proportion:
$$\frac{a+b}{a} = \frac{a}{b} = \varphi$$
A line segment divided into two parts (a and b, where a > b) exhibits the golden ratio when the ratio of the total length to the longer part equals the ratio of the longer part to the shorter part.
Formula and properties
The exact value is:
$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\ldots$$
Key property: φ² = φ + 1 (the only positive number whose square equals itself plus 1):
$$\varphi^2 = \varphi + 1$$
Its reciprocal: 1/φ = φ − 1 ≈ 0.618:
$$\frac{1}{\varphi} = \varphi - 1 \approx 0.6180\ldots$$
Given the longer side a:
$$b = \frac{a}{\varphi}, \quad a + b = a \times \varphi$$
| Known | Longer (a) | Shorter (b) | Total (a+b) |
|---|---|---|---|
| Longer = 100 | 100 | 61.803 | 161.803 |
| Shorter = 100 | 161.803 | 100 | 261.803 |
| Total = 100 | 61.803 | 38.197 | 100 |
| Longer = 1920 (HD width) | 1920 | 1186.6 | 3106.6 |
| Longer = 210 (A4 width mm) | 210 | 129.8 | 339.8 |
Connection to Fibonacci numbers
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…) converges to φ through its consecutive ratios:
$$\frac{F_{n+1}}{F_n} \to \varphi \text{ as } n \to \infty$$
5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, 55/34 = 1.6176. The ratios oscillate above and below φ, converging exponentially fast. This is why φ appears in sunflower seed spirals, nautilus shells, and plant leaf arrangements — all governed by Fibonacci-type growth.
Applications in design and art
φ is used as a proportional guideline in visual design, architecture, and typography — not as a rigid rule, but as a ratio that many people perceive as naturally balanced.
| Domain | Application | Example |
|---|---|---|
| Logo design | Element proportions | Width:height = φ |
| Typography | Body text : heading | 16 px body → 25.9 px heading |
| Photography | Rule of thirds variant | Horizon at 61.8% from edge |
| Architecture | Facade proportions | Height:width = φ |
| Web design | Sidebar:content ratio | 38%:62% column split |
| UI spacing | Padding scale | 8 px, 12.9 px, 20.9 px… |
Note: the claim that the Parthenon, Mona Lisa, or pyramids were designed using φ is disputed by modern scholars — the ratios are approximate and may be coincidental. φ is a useful design guide but not a mystical law.
Frequently asked questions
What is the golden ratio?
φ ≈ 1.6180339887. The unique ratio where (a+b)/a = a/b. Irrational, with φ² = φ + 1.
How do you calculate golden ratio dimensions?
Given longer side a: shorter = a ÷ φ, total = a × φ. Given shorter b: longer = b × φ. Given total: longer = total ÷ φ.
What is the golden ratio in simple terms?
Split a line so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part. That ratio is φ ≈ 1.618.
What is 1/φ?
1/φ = φ − 1 ≈ 0.6180. This means the shorter part is ≈ 61.8% of the longer part.