📐 Golden Ratio Calculator

01. Enter one value and get both sides of the golden ratio

Calculations follow the golden ratio (φ ≈ 1.618), so whichever measurement you already have yields its counterpart immediately.

02. Treat your input as the longer side or the shorter one

Whichever dimension is already fixed, a single toggle switches which side gets calculated.

03. A golden spiral diagram makes the proportion visible

The figure redraws with your results, giving the ratio a shape rather than leaving it as an abstract number.

04. Set how many decimal places you want

Show a rough figure or a precise one depending on whether you're sketching or matching exact dimensions.

05. Copy any result with a single click

Longer side, shorter side and the difference all copy to your clipboard, so nothing gets mistyped into your design tool.

01. Basic Usage

1. Enter the length you want to work from as the base value
2. The longer side (base × φ) and shorter side (base ÷ φ) appear as you type
3. Press the copy button next to the value you need

02. Diagram and Display Options

1. By default the golden spiral diagram draws your input as the longer side
2. Press 'Use shorter side as base' to draw it as the shorter side instead
3. Use the decimal places control to show anywhere from 0 to 6 digits

Q. What does the base-side toggle change?

A. It swaps whether the golden spiral diagram draws your input as the longer or the shorter side. By default the input is the longer side; pressing 'Use shorter side as base' redraws it as the shorter side. Useful when a finished dimension is already fixed. The longer, shorter, and φ values listed on the right are unaffected by this toggle.

Q. What does the golden spiral diagram show?

A. It draws the spiral that emerges when a rectangle built from the longer and shorter sides is repeatedly divided by the golden ratio. The 'difference' label is the gap between the two sides—the strip left over after cutting out a square.

Q. How many decimal places can be shown?

A. From 0 to 6. For pixel-based layouts 0–1 places is usually enough; for print dimensions 2–3 places is a practical range.

Q. What input range is accepted?

A. From 0.001 to 1,000,000. Values outside this range produce no result. No unit is assumed, so you can work in mm, px, cm, or anything else.

Q. Can I copy the value of φ itself?

A. Yes. Alongside the longer and shorter sides, the golden ratio (φ) row has its own copy button, based on 1.618033988749895. φ is always shown to three decimal places, regardless of your decimal place setting.

Q. What is the golden ratio?

A. A ratio of approximately 1.618, known as 'the most beautiful proportion' since ancient Greece. It divides a line so that the whole relates to the longer part exactly as the longer part relates to the shorter one.

Q. Why is it called φ (phi)?

A. Named after the Greek sculptor Phidias, who created the sculptures of the Parthenon—works believed to incorporate the ratio.

Q. What's the difference between the golden ratio and silver ratio?

A. The golden ratio is about 1:1.618, while the silver ratio is about 1:1.414 (√2). The silver ratio is used in A4 paper sizing, where folding in half preserves the aspect ratio.

Q. How does the Fibonacci sequence relate to the golden ratio?

A. Taking the ratio of adjacent numbers in 1, 1, 2, 3, 5, 8, 13... converges on the golden ratio as the numbers grow. 13÷8 = 1.625 and 21÷13 ≈ 1.615, closing in on φ at 1.618.

Q. Tips for using the golden ratio in design?

A. Apply it to layout widths, heights, and margins to create natural, beautiful balance. Using it for a few key divisions works better than forcing every element to match.

01. The Golden Ratio in Nature

The golden ratio appears throughout nature—in sunflower seed arrangements, nautilus shells, and human body proportions. The sunflower is the soundest example: each seed sits about 137.5 degrees around from the last. That angle divides a full turn in the golden ratio and is called the golden angle. Arranged this way, seeds never stack on top of one another, packing the greatest number into a limited area.

02. Use in Design and Art

It's also said to be used in logos by Apple and Twitter. The Parthenon in ancient Greece and the Egyptian pyramids are believed to incorporate the golden ratio, and Leonardo da Vinci's 'Mona Lisa' and 'The Last Supper' show golden ratio compositions. Some research suggests that Beethoven's Symphony No. 5 and Debussy's works also use the golden ratio in their musical structures.

03. Most of Those Stories Came Later

Those examples are widely repeated, yet very few come with evidence that the creator actually intended the golden ratio. The Parthenon's proportions shift depending on which edges you choose to measure; pick the measurement that yields the golden ratio and the golden ratio duly appears. The pyramids are the same. As for the Mona Lisa, no record survives of da Vinci ever mentioning the ratio. He did illustrate De Divina Proportione, a book genuinely about the golden ratio — but illustrating a book is not the same as applying it to one's own painting. Psychology experiments claiming people find the proportion most beautiful have produced inconsistent results. Scholars have argued that the story of the golden ratio as an ancient standard of beauty is itself a nineteenth-century invention.

04. The Fibonacci Connection

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) approaches the golden ratio as numbers get larger. The ratios oscillate as they converge: 3÷2 = 1.5, 5÷3 ≈ 1.667, 8÷5 = 1.6, 13÷8 = 1.625, closing in on 1.618. This tool's diagram uses Fibonacci numbers too, drawing a 34-by-21 rectangle. That gives 34÷21 ≈ 1.619 — an approximation, not φ exactly.

05. The Most Irrational Number

As a continued fraction, φ is written [1; 1, 1, 1, ...] — nothing but ones. The smaller a continued fraction's terms, the harder the number is to approximate with fractions, which makes φ the number least well approximated by any fraction. That property is exactly why the sunflower's golden angle works. Were the rotation angle expressible as a simple fraction, seeds would line up in the same direction after a few turns. Using a number this resistant to approximation keeps them from ever overlapping.

06. Where the Name φ Comes From

The symbol φ came into use only in the early twentieth century; it was not an ancient convention. It is generally traced to the initial of the Greek sculptor Pheidias. Before that, the proportion went by names such as the division ratio or the divine proportion. Even the term 'golden ratio' itself does not appear in the literature until the nineteenth century.

07. The Curiosity of 1.618 and 0.618

φ has an unusual property: subtract one and you get its reciprocal; add one and you get its square. That is, φ − 1 = 1/φ ≈ 0.618 and φ + 1 = φ² ≈ 2.618. The decimals all share the same 618... because φ is a solution to x² = x + 1. Both properties fall straight out of rearranging that single equation.

08. The Base-Switching Button

By default the diagram draws your entered value as the longer side. Pressing the button to switch to the shorter side redraws it treating your value as the shorter side instead. One caveat: only the labels within the diagram change. The longer, shorter, and φ figures listed on the right always show your value times φ and divided by φ, and this button does not affect them. The feature exists to match the diagram to a finished dimension you have already fixed.

09. What the Difference Value Means

Alongside the longer and shorter sides, the diagram shows a difference — the longer side minus the shorter one. Cut a square whose side equals the shorter side out of a golden rectangle, and what remains is another golden rectangle. The width of that remainder is the difference. The operation can be repeated forever, and joining the corners traces a spiral — the very spiral drawn in the diagram.

10. Choosing Decimal Places

You can select from 0 to 6 decimal places. For pixel-based layouts, 0 to 1 works well; for print dimensions, 2 to 3 is a comfortable range. Note that the value of φ itself always displays at three decimal places regardless of your selection. It is a constant rather than a calculated result, so varying its precision serves little purpose.