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.
- φ ≈ 1.6180339887... is irrational - it cannot be expressed as an exact fraction, and its decimal digits never repeat.
- φ² = φ + 1: the only positive number whose square equals itself plus 1. This is what makes it unique among all ratios.
- 1/φ = φ - 1 ≈ 0.618: the reciprocal shares the same decimal part as φ itself, a property no other number has.
- Fibonacci ratios (13/8, 21/13, 34/21...) converge to φ from alternating sides, getting exponentially closer with each step.
- Claims that the Parthenon or Mona Lisa were deliberately designed using φ are disputed by modern scholars - the matching ratios are approximate and likely coincidental.
How to use our golden ratio (phi) 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.
Golden ratio explained to a beginner
Imagine cutting a piece of ribbon into two pieces. If you cut it at just the right spot, the longer piece relates to the shorter piece in exactly the same way the whole ribbon relates to the longer piece. That one special spot - where the ratio repeats itself - is the golden ratio.
The number that makes this work is 1.6180... Say you have a ribbon 161.8 cm long and cut off 100 cm. The longer piece (100 cm) divided by the shorter piece (61.8 cm) = 1.618. The whole ribbon (161.8 cm) divided by the longer piece (100 cm) = 1.618. Both ratios are identical. φ is the only number in the universe where this self-referencing property holds.
Formula and properties for finding golden ratio
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 |
Phi 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.
In practice working with web design and CSS, I find the Fibonacci convergence the most useful design shortcut. Starting from an 8 px spacing base and multiplying by 1.618 each step gives 8, 13, 21, 34 px - which are Fibonacci numbers.
That sequence makes a spacing and padding scale where every step feels proportionally larger without any arbitrary jumps, and it's the reason many design systems already land on those values intuitively.
Phi 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 phi is disputed by modern scholars - the ratios are approximate and may be coincidental. Phi is a useful design guide but not a mystical law.
I've noticed phi is more reliable in typography than in architecture. Setting a heading at 1.618 × the body text size gives a clear visual hierarchy without overshooting. For 16 px body text, that's 25.9 px - very close to the common 24 px or 28 px step.
Those sizes feel natural to most designers precisely because they sit near phi (φ), even when nobody calculated it deliberately.
FAQs about golden ratio and phi
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.
Why is φ called phi?
The Greek letter φ (phi) was adopted in honour of the sculptor Phidias, who is associated with the Parthenon. The name "golden ratio" traces to Luca Pacioli's Divina Proportione (1509), which described the proportion as having divine aesthetic properties. The term became widespread in 19th-century mathematical literature.
Is φ truly irrational?
Yes. The exact value is (1 + √5)/2. Since √5 cannot be expressed as a ratio of two integers, neither can φ. Its decimal expansion is infinite and non-repeating, which means no fraction represents it exactly - only approximations like 1.618 or 89/55.
What is a golden rectangle?
A rectangle whose length-to-width ratio equals φ. If you remove the largest square from one end, the remaining piece is another golden rectangle in the same proportion. This self-similarity repeats infinitely and produces a logarithmic spiral - the shape commonly associated with nautilus shells.
Quiz: how well do you know the golden ratio?
1. Which identity uniquely defines φ among all positive numbers?
2. From the reference table, if the total length (a+b) is 100, what is the longer part (a)?
3. What is 1/φ approximately equal to?
4. According to the applications table, what heading size does φ suggest for 16 px body text?
5. What does the page say about claims that the Parthenon was designed using φ?