Golden Ratio Calculator
Enter the total, the longer segment or the shorter segment to find the other two in the golden ratio, φ ≈ 1.618.
Updated
Golden Ratio Calculator
Our Golden Ratio Calculator splits a length into two parts in the golden ratio, or finds the missing lengths when you know one part. Enter the total, the longer segment or the shorter segment, and it returns the other two straight away, with the formula used and a check that the parts are in the ratio φ ≈ 1.618. It works in any unit: centimeters, inches, pixels or anything else.
Quick answer
The golden ratio is φ = (1 + √5) ÷ 2 ≈ 1.6180339887. A length is split in the golden ratio when the whole divided by the longer part equals the longer part divided by the shorter part. For a total of 100, the longer part is 100 ÷ 1.618 ≈ 61.803 and the shorter part is ≈ 38.197.
- Longer part: total × 0.618 (total ÷ φ)
- Shorter part: total × 0.382 (total ÷ φ²)
- From the longer part: shorter = a ÷ φ, total = a × φ
- From the shorter part: longer = b × φ, total = b × φ²
What Is the Golden Ratio?
Cut a line into a longer part a and a shorter part b. The cut is in the golden ratio when the whole line is to the longer part as the longer part is to the shorter part:
φ = (1 + √5) ÷ 2
≈ 1.6180339887
φ is the Greek letter phi. It is an irrational number, so its decimals never end or repeat. The longer part is always about 61.8% of the whole and the shorter part about 38.2%.
Golden Ratio Formulas
| You know | Longer (a) | Shorter (b) |
|---|---|---|
| Total T | T ÷ φ | T ÷ φ² |
| Longer a | a | a ÷ φ |
| Shorter b | b × φ | b |
Useful constants: 1 ÷ φ = φ − 1 ≈ 0.6180339887, 1 ÷ φ² = 2 − φ ≈ 0.3819660113 and φ² = φ + 1 ≈ 2.6180339887. The total is a + b in every case.
Where 1.618 Comes From
Take the shorter part as 1 and the longer part as φ. The rule (a + b) ÷ a = a ÷ b becomes:
φ² = φ + 1
φ² − φ − 1 = 0
φ = (1 + √5) ÷ 2
The quadratic formula gives two roots; the positive one is φ ≈ 1.618. This also explains the neat facts above: squaring φ adds 1, and dividing 1 by φ takes 1 away.
Worked Examples
Split 100 cm.
≈ 61.80339887 cm
b = 100 − a
≈ 38.19660113 cm
The longer part is 10. b = 10 ÷ φ ≈ 6.180339887, and the total is 10 × φ ≈ 16.18033989.
The shorter part is 5. a = 5 × φ ≈ 8.090169944, and the total is 5 × φ² ≈ 13.09016994.
A 960 px layout. A main column of 960 ÷ φ ≈ 593.3 px and a sidebar of ≈ 366.7 px split the width in the golden ratio.
Golden Ratio Table
| Total | Longer (a) | Shorter (b) |
|---|---|---|
| 1 | 0.618034 | 0.381966 |
| 10 | 6.18034 | 3.81966 |
| 12 | 7.41641 | 4.58359 |
| 100 | 61.8034 | 38.1966 |
| 960 | 593.313 | 366.687 |
| 1,000 | 618.034 | 381.966 |
| 1,920 | 1,186.63 | 733.375 |
Fibonacci and the Golden Ratio
In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, each number divided by the one before gets closer and closer to φ:
| Ratio | Value |
|---|---|
| 5 ÷ 3 | 1.666667 |
| 8 ÷ 5 | 1.6 |
| 13 ÷ 8 | 1.625 |
| 21 ÷ 13 | 1.615385 |
| 34 ÷ 21 | 1.619048 |
| 89 ÷ 55 | 1.618182 |
| 144 ÷ 89 | 1.617978 |
That is why splits like 5:3, 8:5 and 13:8 are used as quick whole number stand-ins for the golden ratio.
The Golden Ratio in Design
- Layouts: split a page width into a main column and a sidebar, about 62% and 38%.
- Typography: multiply a 16 px body size by φ for a heading of about 26 px, and again for about 42 px.
- Golden rectangle: sides in the ratio 1 : 1.618. Cutting a square off it leaves a smaller golden rectangle.
- Photography: the phi grid places lines at 38.2% and 61.8% of the frame, a little closer to the middle than the rule of thirds.
- Logos and products: proportions of shapes and spacing.
The golden ratio does appear in nature, for example in the spiral arrangement of sunflower seeds. Many popular claims about famous buildings and paintings, though, rest on loose measurements and are disputed.
Common Mistakes
- Multiplying the total by φ. The parts are smaller than the total, so divide: 100 ÷ φ ≈ 61.8, not 161.8.
- Mixing up a and b. a is always the longer part, about 61.8% of the whole.
- Rounding φ to 1.6. That is 8 ÷ 5, not the golden ratio; use 1.618 or more digits.
- Splitting in half first. The golden cut is at 61.8% of the length, not 50%.
Key Terms
- φ (phi)
- The golden ratio, (1 + √5) ÷ 2 ≈ 1.618.
- Longer segment
- The bigger part a, about 61.8% of the total.
- Shorter segment
- The smaller part b, about 38.2% of the total.
- Golden section
- A cut that divides a line in the golden ratio.
- Golden rectangle
- A rectangle whose sides are in the ratio 1 : φ.
- Irrational
- A number that cannot be written as a fraction.
How to Use the Golden Ratio Calculator
- Choose what you know. Pick Total if you have the whole length to split, Longer if you know the bigger part a, or Shorter if you know the smaller part b.
- Enter the length. Type a positive number in any unit, such as 100, 12.5 or 2e3. Commas such as 1,920 are fine.
- Read and copy the result. The first missing length appears as you type, with a + b = total underneath. The table lists the total, the longer part and the shorter part, the formula used and the check a ÷ b = φ. Press Copy to copy it, or Clear or Esc to start again.
Example: choose Total and enter 100. The longer segment is 61.80339887, with the working 61.80339887 + 38.19660113 = 100, and the table lists the shorter segment 38.19660113 and the formula a = 100 ÷ φ.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is the golden ratio?
The golden ratio φ (phi) is (1 + √5) ÷ 2 ≈ 1.6180339887. Two lengths a and b, with a the longer, are in the golden ratio when (a + b) ÷ a = a ÷ b = φ.
How do you calculate the golden ratio of a number?
Divide the number by 1.618 to get the longer part, and subtract that from the number to get the shorter part. For 100: 100 ÷ 1.618 ≈ 61.80 and 100 − 61.80 ≈ 38.20.
What is the golden ratio formula?
φ = (1 + √5) ÷ 2, the positive solution of φ² = φ + 1. For a total T, the longer part is T ÷ φ and the shorter part is T ÷ φ². From the longer part a, the shorter part is a ÷ φ; from the shorter part b, the longer part is b × φ.
What is the golden ratio of 100?
100 splits into a longer part of about 61.803 and a shorter part of about 38.197. Multiplying instead, 100 × φ ≈ 161.803 is the total when 100 is the longer part.
Why is the golden ratio 1.618?
Setting the shorter part to 1 and the longer part to φ, the rule (a + b) ÷ a = a ÷ b becomes φ² = φ + 1. Solving gives φ = (1 + √5) ÷ 2 ≈ 1.618.
What is 0.618 in the golden ratio?
0.618 is 1 ÷ φ, which also equals φ − 1. Multiplying a length by 0.618 gives its longer golden part, and multiplying by 0.382 (1 ÷ φ²) gives the shorter part.
How is the golden ratio related to the Fibonacci sequence?
Dividing each Fibonacci number by the one before it gives ratios that get closer and closer to φ: 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, 21 ÷ 13 ≈ 1.6154, 144 ÷ 89 ≈ 1.6180.
How do you use the golden ratio in design?
Split widths about 62% and 38%, for example a 960 px layout into a 593 px main column and a 367 px sidebar, or scale font sizes by 1.618, such as 16 px body text with 26 px headings.
What is a golden rectangle?
A rectangle whose long side is φ times its short side, such as 10 by 16.18. Cutting a square off one end leaves a smaller rectangle with the same golden proportions.
Is the Golden Ratio Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Golden Ratio Calculator to split any length into golden ratio parts in an instant.