Golden Ratio Calculator (φ)

Split any length or rectangle into golden sections, compute φ-based proportions, and explore how to use the golden ratio in math, art, and design.

Units are optional (cm, px, in, etc.).

Visual guide: longer ≈ 61.8%, shorter ≈ 38.2% of total

What is the golden ratio?

The golden ratio, usually written as phi (φ), is an irrational number that appears in geometry, art, architecture, and design. Its value is:

Exact value:

\[ \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\ldots \]

A line is divided in the golden ratio when the whole length divided by the longer part is equal to the longer part divided by the shorter part.

Golden ratio formulas

1. Definition using a line segment

Suppose a line of total length \(L\) is split into a longer part \(a\) and a shorter part \(b\), with \(a > b > 0\). The golden ratio condition is:

\[ \frac{a + b}{a} = \frac{a}{b} = \varphi \]

From this, we can derive the classic quadratic equation:

\[ \varphi = 1 + \frac{1}{\varphi} \quad\Rightarrow\quad \varphi^2 - \varphi - 1 = 0 \]

2. Split a length into golden sections

Given the total length \(L = a + b\):

\[ a = \frac{\varphi}{1 + \varphi} \, L \approx 0.618\,L \]

\[ b = \frac{1}{1 + \varphi} \, L \approx 0.382\,L \]

Our calculator uses these formulas to compute the longer and shorter segments for any length you enter.

3. Golden rectangle

A golden rectangle has side lengths in the ratio \(\varphi : 1\). If you know one side, the other is:

Given height \(h\):

\[ \text{width} = \varphi \, h \]

Given width \(w\):

\[ \text{height} = \frac{w}{\varphi} \]

How to use this golden ratio calculator

1) Split a length (for grids, margins, and spacing)

  1. Enter the total length (e.g., 1200 px, 30 cm, 8 in).
  2. Optionally type a unit label (px, cm, in, etc.).
  3. Click “Calculate golden split”.
  4. The tool returns:
    • the longer segment (≈ 61.8% of the total)
    • the shorter segment (≈ 38.2% of the total)
    • the ratio \(a : b\) compared to φ

Use this for column widths, sidebar vs. content, or spacing between elements.

2) Golden rectangle (for layouts and cropping)

  1. Choose whether you know the width or height.
  2. Enter the value and unit (e.g., 1080 px height).
  3. Click “Calculate rectangle”.
  4. The calculator outputs the matching side and the exact aspect ratio.

This is useful for designing artboards, posters, slides, or cropping photos to a golden rectangle.

3) Check a ratio (how “golden” is it?)

  1. Enter two positive values \(a\) and \(b\) (with \(a\) intended as the larger).
  2. Click “Check golden-ness”.
  3. The tool computes:
    • \(a / b\)
    • the percentage difference from φ
    • a simple verdict (very close / close / not close)

This is handy to test existing designs, photo crops, or real-world measurements against the golden ratio.

Examples

Example 1 – Web layout width

Suppose your content area is 1200 px wide and you want a golden split between main content and sidebar.

  • Total \(L = 1200\) px.
  • Longer segment \(a \approx 0.618 \times 1200 = 741.6\) px.
  • Shorter segment \(b \approx 0.382 \times 1200 = 458.4\) px.

You might round to 742 px main content and 458 px sidebar.

Example 2 – Golden rectangle from height

You want a poster with height 70 cm in golden ratio.

  • Given height \(h = 70\) cm.
  • Width \(w = \varphi \, h \approx 1.618 \times 70 \approx 113.26\) cm.

So a golden rectangle poster would be about 113 × 70 cm.

Example 3 – Is my photo crop golden?

Your image is 1920 × 1080 px. Check the ratio:

  • \(a = 1920\), \(b = 1080\)
  • \(a / b = 1920 / 1080 \approx 1.7778\)
  • Difference from φ ≈ \(|1.7778 - 1.6180| / 1.6180 \approx 9.9\%\)

So 16:9 is not particularly close to the golden ratio.

Golden ratio in math, nature, and design

1. Fibonacci numbers and φ

The golden ratio is closely related to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …). The ratio of consecutive Fibonacci numbers approaches φ:

\[ \frac{F_{n+1}}{F_n} \to \varphi \quad \text{as } n \to \infty \]

This connection explains why φ appears in spirals and growth patterns that can be approximated by Fibonacci-like progressions.

2. Golden ratio in nature

Many natural patterns can be approximated by golden-ratio-related spirals or proportions, such as:

  • phyllotaxis (arrangement of leaves and seeds)
  • some flower petal counts (often Fibonacci numbers)
  • spiral shells and hurricanes (logarithmic spirals close to φ-based growth)

These are not perfect golden ratios, but φ often provides a useful mathematical model.

3. Golden ratio in art, architecture, and UX

Historically, the golden ratio has been associated with:

  • classical architecture and Renaissance paintings
  • book page layouts and typographic grids
  • logo design and icon proportions
  • modern UI/UX layouts (e.g., content vs. sidebar, hero vs. body)

While some historical claims are exaggerated, φ remains a practical guideline for creating balanced compositions.

Practical tips for designers

  • Start with content: Use φ to refine spacing and hierarchy, not to force content into awkward shapes.
  • Combine with other systems: Golden ratio works well alongside grids, 8‑pt spacing, and rule-of-thirds.
  • Use ranges, not absolutes: Being within a few percent of φ is usually visually indistinguishable from exact φ.
  • Test with users: A mathematically “perfect” layout can still fail usability tests—prioritize clarity and accessibility.

FAQ

What is the golden ratio?

The golden ratio is the number \(\varphi \approx 1.618\) that describes a special proportional relationship: the whole is to the larger part as the larger part is to the smaller part.

What is the exact formula for φ?

φ is the positive solution of the quadratic equation \(\varphi^2 - \varphi - 1 = 0\), so:

\[ \varphi = \frac{1 + \sqrt{5}}{2} \]

How accurate do I need to be?

For visual design, being within 1–2% of φ is usually more than enough. Our ratio checker shows the percentage difference so you can judge how close you are.

Can I use any units?

Yes. The golden ratio is dimensionless, so you can use pixels, centimeters, inches, points, or any other unit. Just be consistent within a calculation.

Is the golden ratio scientifically “proven” to be most beautiful?

Not universally. Some studies find preferences for φ-like proportions in specific contexts, but results are mixed. The golden ratio is best treated as a useful design heuristic, not a universal law of beauty.