Golden Ratio Calculator — Split a Length into a : b = φ

How the Golden Ratio Split Works

The golden ratio, φ = (1 + √5) / 2 ≈ 1.61803, appears when a line is divided into two parts a (short) and b (long) so that b / a = L / b = φ, with L = a + b.

• Given a: b = φa, L = φ²a.
• Given b: a = b / φ, L = φb.
• Given L: b = L / φ, a = L / φ².

The diagram shows a golden rectangle of sides b (long) and a (short), partitioned into a square of side a and a smaller golden rectangle—this self-similarity is a hallmark of φ.

Golden Ratio Calculator: FAQs

What is the golden ratio φ?

φ (phi) = (1+√5)/2 ≈ 1.61803. In a golden split, b/a = φ and L/b = φ.

Which inputs are valid?

Any one of: short part a, long part b, or total length L. Entering two or more is fine; the tool checks consistency.

What formulas do you use?

Given a → b=φa, L=φ²a; given b → a=b/φ, L=φb; given L → b=L/φ, a=L/φ².

Is my data private?

Yes. Everything runs locally in your browser; nothing is uploaded.

What Is the Golden Ratio (φ)?

The Golden Ratio, usually written as φ (phi), is the constant φ = (1 + √5) / 2 ≈ 1.61803. It appears when a length is split into a short part a and a long part b such that the proportions match: b / a = (a + b) / b = φ. Rearranging leads to the classic quadratic φ² = φ + 1, which uniquely characterizes phi. A rectangle whose sides have ratio b : a = φ : 1 is called a golden rectangle; removing the largest possible square leaves a smaller, similar rectangle, revealing φ’s self-similarity.

Formulas You’ll Use Often

• Given the short part a: b = φ·a, L = a + b = φ²·a.
• Given the long part b: a = b / φ, L = φ·b.
• Given the total L: b = L / φ, a = L / φ².
• Golden rectangle sizes: width = b, height = a (or the inverse if rotated).

Connections to the Fibonacci Sequence

The ratio of consecutive Fibonacci numbers approaches φ: 1/1, 2/1, 3/2, 5/3, 8/5, … → 1.618…. This happens because the Fibonacci recurrence Fn+1 = Fn + Fn-1 leads to the same characteristic equation as the definition of φ. Designers sometimes approximate φ with nearby “Fibonacci ratios” like 8:5 or 13:8 for grids, spacing, or typographic scales.

Why Designers and Builders Care

The golden ratio is valued for balanced proportions. In practice, it serves as a helpful guideline for layout, cropping, and hierarchy rather than a strict rule. Examples include:

• Layout & UI: Split a canvas into 1 : φ columns, or size a sidebar to 1/φ of the total width.
• Type & spacing: Scale font sizes, margins, or image gutters by multiplying/dividing by φ.
• Product & print: Choose aspect ratios near 1.618:1 for cards, posters, and hero images.

Common Misconceptions

Not every “pleasing” object or famous artwork was intentionally designed with φ. Historical claims are often overstated or retrofitted. The golden ratio can be useful, but it isn’t a magic key to beauty— other ratios (thirds, root-2, root-3, or simple halves) also produce excellent results depending on the context. Treat φ as one option in a toolkit, and test with real users.

Quick Construction Tips

• To split a length L into golden parts: b = L / φ and a = L - b = L / φ².
• To build a golden rectangle from a square of side a: extend one side to length b = φ·a.
• For grids: start with a base unit u and scale by powers of φ (e.g., u, φu, φ²u).

Tip: In our calculator, enter any one of a, b, or L, choose decimal places and units, and the other values update instantly—everything runs locally in your browser.