Golden Ratio Calculator | Phi, Golden Section, Golden Mean

Q: What is the golden ratio?

The golden ratio is the mathematical constant φ = 1.6180339887... . It describes the proportion where (a + b) / a = a / b . It is also called the golden mean , golden section , or divine proportion .

Q: What does this golden ratio calculator find?

This calculator finds the longer part a , the shorter part b , or the total length a + b for a golden section. Enter one known value and the other two are calculated automatically.

Q: What is the golden ratio formula?

The main formula is φ = (a + b) / a = a / b . From this, the common working forms are a = b × φ , b = a / φ , a + b = a × φ , and a + b = b × φ² .

Q: Example: if the longer part is 10, what are the shorter part and the total?

If a = 10 , then b = 10 / φ ≈ 6.1803 and a + b = 10 × φ ≈ 16.1803 .

Golden ratio calculator for phi (φ = 1.6180339887...). Calculate the longer part, shorter part, or total length in the golden section using the formulas φ = (a+b)/a = a/b.

Use this golden ratio calculator to split a length into the golden section or to find the missing long part, short part, or total length when the proportion follows phi (φ). It is useful for geometry, design, art, photography, layout work, and proportion checks.

The golden ratio is defined by φ = (a + b) / a = a / b, where a is the longer part, b is the shorter part, and a + b is the total length. This calculator lets you enter any one of those lengths and solve the other two in the same proportion.

\begin{align} \phi &= \frac{a+b}{a}=\frac{a}{b}\\ \\ \phi &= 1.6180339887498948482... \end{align}

A (longer) =

B (shorter) =

A+B =

Golden Ratio FAQs

What is the golden ratio?
The golden ratio is the mathematical constant φ = 1.6180339887.... It describes the proportion where (a + b) / a = a / b. It is also called the golden mean, golden section, or divine proportion.

What does this golden ratio calculator find?
This calculator finds the longer part a, the shorter part b, or the total length a + b for a golden section. Enter one known value and the other two are calculated automatically.

What is the golden ratio formula?
The main formula is φ = (a + b) / a = a / b. From this, the common working forms are a = b × φ, b = a / φ, a + b = a × φ, and a + b = b × φ².

How do I divide a line in the golden ratio?
If the total length is known, divide it so that the longer part satisfies a = total / φ and the shorter part satisfies b = total / φ². The ratio of a to b will then equal φ.

Example: if the longer part is 10, what are the shorter part and the total?
If a = 10, then b = 10 / φ ≈ 6.1803 and a + b = 10 × φ ≈ 16.1803.

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Golden Ratio FAQs