Quick answer
The golden ratio is φ = (1 + √5) ÷ 2 ≈ 1.618034. A longer part of 1.618034 pairs with a shorter part of 1, for a total of 2.618034.
Enter the longer part. The calculator returns the shorter part and the total.
Related tools: Ratio Calculator: Simplify a ratio of two whole numbers. Area Calculator: Area of common two-dimensional shapes. Right Triangle Calculator: Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
How to use the Golden Ratio Calculator
- Enter longer part (a) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Longer part (a) | A number | > 0 |
Formula and method
Formula
φ = (1 + √5) ÷ 2 ≈ 1.618034; b = a ÷ φ; total = a + b
IEEE-754 double precision; shown to at most 6 decimal places.
Given the longer part, the shorter part is a ÷ φ.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Longer part (a) = 1.618033988749895.
| Result | Value |
|---|---|
| Shorter part (b) | 1 |
| Total | 2.618034 |
Example 2
Inputs: Longer part (a) = 10.
| Result | Value |
|---|---|
| Shorter part (b) | 6.18034 |
| Total | 16.18034 |
The first line is the headline result (shorter part (b)). Below it the golden Ratio Calculator lists total, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Definition
Two quantities are in the golden ratio if (a + b) ÷ a = a ÷ b = φ. So b = a ÷ φ, and the total is a + b = a × (1 + 1/φ) = 1.618a. Here the longer part is 1.618, the shorter part is 1 and the total is 2.618.
In practice
Designers use φ for layouts and proportions, and it is the limit of the ratio of consecutive Fibonacci numbers. Claims that it appears everywhere in nature and art are often overstated.
How the result changes with longer part
Holding every other input at the first worked example, this table changes only longer part. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Longer part | Shorter part (b) | Total |
|---|---|---|
| 0.809 | 0.499989 | 1.308989 |
| 1.618 | 0.999979 | 2.617979 |
| 3.236 | 1.999958 | 5.235958 |
| 6.472 | 3.999916 | 10.471916 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Longer part (a): entering "0" shows "Longer part (a) must be greater than 0." beside that field and no result.
Choosing between the Golden Ratio Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Pythagorean Theorem Calculator](/math/pythagorean-theorem-calculator/) | find, leg a and other side | Hypotenuse c |
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Positive length
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for longer part (a) to confirm this.
Frequently asked questions
What is 1/φ?
0.618034, which is φ − 1.
What is the golden rectangle?
One whose sides are in ratio φ.
Is φ irrational?
Yes. φ = (1 + √5) ÷ 2 is irrational because √5 is.
What does the Golden Ratio Calculator calculate?
Split a length into golden-ratio parts. It returns Shorter part (b) and Total.
What formula does the Golden Ratio Calculator use?
φ = (1 + √5) ÷ 2 ≈ 1.618034; b = a ÷ φ; total = a + b. Given the longer part, the shorter part is a ÷ φ.
What do I need to enter in the golden Ratio Calculator?
You enter Longer part (a).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Positive length.
What is an example of a golden Ratio result?
With longer part (a) 1.618033988749895, the shorter part (b) is 1.
Where does the method come from?
The method follows: Euclid's Elements, Book VI, Prop. 30 (extreme and mean ratio).
Related calculators
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Area Calculator — Area of common two-dimensional shapes.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Pythagorean Theorem Calculator — Find the hypotenuse or a missing leg of a right triangle.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
Sources
- Euclid's Elements, Book VI, Prop. 30 (extreme and mean ratio)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.