Golden Ratio Calculator
The golden ratio, φ ≈ 1.618, splits a length so that the longer part divided by the shorter equals the whole divided by the longer part. A length of 100 splits into about 61.8 and 38.2 (see the split). Use the calculator to split any length, size a golden rectangle, check a ratio, or build a type scale.
Find golden ratio proportions
Type one length and say which part it is. The other two parts come out in the golden ratio.
Either order works. You can also type a size such as 1920x1080 in the first field.
Usually your body text size. The scale runs three steps down and five up.
Check your input
The split
Golden rectangle and spiral
Among common shapes
Longer side ÷ shorter side for squares, photos, paper and screens. The blue band is φ to within 1%.
Type scale
Each size is φ times the one below it. The highlighted row is your base.
How to Use This Calculator
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Pick a job
Split a length, size a golden rectangle, check a ratio, or build a design scale.
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Type one size
Say which part you know: the total, the longer part or the shorter part. Add a unit if you like.
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Read and copy
You get every matching size, the exact form, and a drawing of the rectangle, spiral or grid.
What is the golden ratio?
Cut a line into two parts, a longer part a and a shorter part b. The cut is golden when the whole line is to a as a is to b. That shared ratio is the golden ratio, written φ (the Greek letter phi):
It is also called the golden section, the golden mean or the divine proportion. Euclid described it around 300 BC as cutting a line in "extreme and mean ratio". Like π, φ is irrational: its decimals never end or repeat, so every length in a golden split is rounded somewhere.
Where 1.618 comes from
Set the shorter part to 1 and the longer part to φ. Then (φ + 1) ÷ φ = φ, which rearranges to φ² = φ + 1, or φ² − φ − 1 = 0. The quadratic formula gives φ = (1 + √5) ÷ 2. The other solution, (1 − √5) ÷ 2 ≈ −0.618, is negative, so it cannot be a length. That equation gives φ some unusual properties:
| Expression | Same as | Value |
|---|---|---|
| φ | (1 + √5) ÷ 2 | ≈ 1.618034 |
| φ² | φ + 1 | ≈ 2.618034 |
| 1 ÷ φ | φ − 1 | ≈ 0.618034 |
| 1 ÷ φ² | 2 − φ | ≈ 0.381966 |
Squaring φ adds 1, and dividing by it subtracts 1. The last two rows are the 61.8% and 38.2% you see everywhere: the longer part of a golden split is 61.8% of the whole, the shorter part 38.2%.
How to split a length in the golden ratio
Divide the total by φ for the longer part, and subtract to get the shorter part. If you know a part instead, multiply or divide by φ to move between them: b × φ = a, and a × φ = the total. Here are some ready-made splits:
| Total | Longer part a | Shorter part b |
|---|---|---|
| 1 | ≈ 0.618 | ≈ 0.382 |
| 10 | ≈ 6.1803 | ≈ 3.8197 |
| 12 | ≈ 7.4164 | ≈ 4.5836 |
| 100 | ≈ 61.8034 | ≈ 38.1966 |
| 1,000 | ≈ 618.034 | ≈ 381.966 |
The golden rectangle and the golden spiral
A golden rectangle has its long side φ times its short side. Cut the largest square off one end and the strip that remains is again a golden rectangle, only φ times smaller. Keep cutting and you get a whirlpool of squares. Draw a quarter circle in each square and you get the golden spiral, which grows by a factor of φ every quarter turn. The calculator draws both for your size.
| Long side | Short side |
|---|---|
| 100 | ≈ 61.8 |
| 960 px | ≈ 593.31 px |
| 1,200 px | ≈ 741.64 px |
| 1,920 px | ≈ 1,186.63 px |
| 11 in | ≈ 6.8 in |
| 30 cm | ≈ 18.54 cm |
Few everyday rectangles are golden. A 16:10 laptop screen and a bank card come close; A4 paper and HD video are well off:
| Shape | Size | Ratio | vs φ |
|---|---|---|---|
| Square | 1 : 1 | 1 | −38.2% |
| Older TV | 4 : 3 | ≈ 1.333 | −17.6% |
| A4 paper | 297 × 210 mm | ≈ 1.414 | −12.6% |
| 35 mm photo | 3 : 2 | 1.5 | −7.3% |
| Bank card | 85.60 × 53.98 mm | ≈ 1.586 | −2% |
| 16:10 laptop screen | 16 : 10 | 1.6 | −1.1% |
| Golden rectangle | φ : 1 | ≈ 1.618 | 0% |
| HD video | 16 : 9 | ≈ 1.778 | +9.9% |
Check the bank card in the calculator
The Fibonacci sequence and φ
In the Fibonacci sequence each number is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21 and so on. Divide each number by the one before it and the ratios swing above and below φ, closing in a little more every step:
| Division | Ratio | Off from φ |
|---|---|---|
| 1 ÷ 1 | 1 | −0.6180 |
| 2 ÷ 1 | 2 | +0.3820 |
| 3 ÷ 2 | 1.5 | −0.1180 |
| 5 ÷ 3 | ≈ 1.666667 | +0.0486 |
| 8 ÷ 5 | 1.6 | −0.0180 |
| 13 ÷ 8 | 1.625 | +0.0070 |
| 21 ÷ 13 | ≈ 1.615385 | −0.0026 |
| 34 ÷ 21 | ≈ 1.619048 | +0.0010 |
| 55 ÷ 34 | ≈ 1.617647 | −0.0004 |
| 89 ÷ 55 | ≈ 1.618182 | +0.0001 |
| 144 ÷ 89 | ≈ 1.617978 | −0.0000565 |
| 233 ÷ 144 | ≈ 1.618056 | +0.0000216 |
| 377 ÷ 233 | ≈ 1.618026 | −0.0000082 |
| 610 ÷ 377 | ≈ 1.618037 | +0.0000031 |
That is why Fibonacci numbers make good stand-ins for φ in practice: 8 × 5, 13 × 8 and 89 × 55 rectangles are all very nearly golden.
Check 89 and 55 in the calculator
The golden ratio in design and photography
Type scale
Multiply a base font size by φ for each larger level and divide by φ for smaller text. From a 16 px body size:
Many designers also use φ for line height, about 1.6 times the font size. The steps get large quickly, so long pages often use a gentler ratio such as 1.25 for headings and keep φ for the biggest jumps.
Layouts
Split a page width in the golden ratio for a main column and a sidebar. A 960 px layout gives about 593 px of content and 367 px of sidebar; a 1200 px one gives about 742 and 458.
Phi grid and the rule of thirds
The phi grid divides a photo with lines 38.2% and 61.8% of the way across and down, so its four crossing points sit a little closer to the center than the rule of thirds (33.3% and 66.7%). On a 6000 × 4000 photo the phi lines fall at about 2,292 and 3,708 px across, against 2,000 and 4,000 for thirds. The calculator draws both on any frame size in Check a ratio mode. Neither rule makes a photo good on its own; they are starting points for placing a subject.
Draw the phi grid on a 6000 × 4000 photo
Myths about the golden ratio
The golden ratio is often said to be hidden in the Parthenon, the Great Pyramid, the Mona Lisa and the human face. Most of these claims do not hold up. In 1992 the mathematician George Markowsky checked claims about buildings such as the Parthenon and the Great Pyramid, and about art, and found them false or seriously misleading: measurements were chosen to fit, and ratios only roughly near 1.6 were called golden. Face proportions are a good example. A study in Vision Research found that people preferred faces whose proportions match an average face, which are not golden.
Sources
- The College Mathematics Journal: Markowsky, G. (1992). Misconceptions about the Golden Ratio, volume 23, issue 1, pages 2 to 19.
- Vision Research: Pallett, P. M., Link, S. and Lee, K. (2010). New “golden” ratios for facial beauty, volume 50, issue 2, pages 149 to 154.
Frequently Asked Questions
What is the golden ratio?
The golden ratio, φ (phi), is about 1.618. Two lengths a and b are in the golden ratio when the longer one divided by the shorter equals the total divided by the longer one: a ÷ b = (a + b) ÷ a. Exactly, φ = (1 + √5) ÷ 2 = 1.6180339887…
How do you calculate the golden ratio of a number?
Divide it by 1.618 for the longer part, then subtract that from the number for the shorter part. For 100: 100 ÷ 1.618 ≈ 61.8 and 100 − 61.8 = 38.2. To go the other way, multiply by 1.618.
What is the golden ratio formula?
φ = (1 + √5) ÷ 2. It comes from a ÷ b = (a + b) ÷ a: write φ for a ÷ b, and the equation becomes φ = 1 + 1/φ, or φ² = φ + 1. The positive solution of that quadratic is (1 + √5) ÷ 2 ≈ 1.618.
What is a golden rectangle?
A rectangle whose long side is φ ≈ 1.618 times its short side, such as 100 × 61.8 or 1920 × 1,187 px. Cut a square off it and what is left is another golden rectangle, which is how the golden spiral is drawn.
How is the golden ratio related to the Fibonacci sequence?
Divide each Fibonacci number by the one before it and the answers close in on φ: 3 ÷ 2 = 1.5, 8 ÷ 5 = 1.6, 21 ÷ 13 ≈ 1.6154, 89 ÷ 55 ≈ 1.6182. The bigger the numbers, the closer the ratio gets to 1.618034.
Is a golden ratio face more attractive?
Not according to the research. Face calculators measure distances such as eyes to mouth against 1.618, but a 2010 study in Vision Research found that faces looked best when the eyes-to-mouth distance was about 36% of the face length and the gap between the eyes about 46% of its width. Those are the proportions of an average face, not the golden ratio.
How do you use the golden ratio in design?
Use φ to pick sizes that relate to each other: a 16 px body font gives about 25.9 px subheadings and 41.9 px headings, and a 960 px page splits into about 593 px of content and 367 px of sidebar. It is a handy rule for consistent proportions, not a guarantee of beauty.