Golden Ratio Calculator

What this calculator does

So, you have a number and you want to see it scaled by the golden ratio, both up and down. This tool builds that scale. Enter a number and it produces a ladder of values around it, five steps larger and five steps smaller, each rung related to the next by the golden ratio, shown both as an exact decimal and as the nearest whole number.

One input, and a full golden-ratio scale centred on your number comes back.

How to use it

1. Enter your number.
2. Press Calculate to see the scale above and below it.

What the golden ratio is

The golden ratio is a special number, roughly 1.618, that has fascinated mathematicians and artists for thousands of years. It is often written with the Greek letter phi. Two lengths are in the golden ratio when the ratio of the larger to the smaller equals the ratio of their sum to the larger. Euclid described it around 300 BC as dividing a line in extreme and mean ratio. The golden ratio is an irrational number, so its decimals run on forever without repeating, and this tool uses the value 1.618034 to scale numbers up, along with its reciprocal, 0.618034, to scale them down.

Building a golden-ratio scale

What the tool produces is a golden-ratio scale, a sequence of numbers where each step is about 1.618 times the one before it. Starting from your number, it climbs upward five times, multiplying by the golden ratio at each step, and descends five times, dividing by the golden ratio at each step. Your original number sits in the middle. The result is a run of values spaced by this famous proportion, with the exact decimal shown alongside the nearest whole number. Scales like this are exactly what designers reach for when they want sizes that feel harmonious rather than arbitrary.

The add-the-last-two trick

There is a neat shortcut hiding in how the scale is built, and it comes from a defining property of the golden ratio: the golden ratio squared equals the golden ratio plus one. That single fact means you can climb the scale without ever multiplying. To get the next term going up, you just add the previous two terms together, and to step down, you subtract. This is the very same rule that drives the Fibonacci sequence, which is no coincidence, since the ratios of consecutive Fibonacci numbers close in on the golden ratio. The list of Fibonacci numbers tool shows that sequence in full.

A worked example

Enter 10. Stepping up, 10 times the golden ratio is about 16.18, and the next step up is about 26.18. Stepping down, 10 divided by the golden ratio is about 6.18, and the step below that is about 3.82. So around your 10 the scale reads roughly 3.82, 6.18, 10, 16.18, 26.18, each about 1.618 times its neighbour. Rounded to the nearest whole numbers, that is 4, 6, 10, 16, 26.

Where the golden ratio is used

The golden ratio shows up in a remarkable range of places. In design and typography it gives type scales and layout proportions that many people find pleasing, which is much of what this tool is handy for. In art and architecture it appears in the golden rectangle, a shape whose sides are in golden proportion. In nature it governs the spirals of shells, sunflowers, and pinecones, tied to the Fibonacci numbers. And in pure mathematics it turns up in the pentagon, in continued fractions, and in the Euclidean algorithm. Few numbers reach across so many different fields from a single simple definition.

Questions people ask

What is the golden ratio?

A special proportion, roughly 1.618, written as the Greek letter phi, where the larger part relates to the smaller as their sum relates to the larger.

What does this tool produce?

A golden-ratio scale: your number stepped up and down by the golden ratio, five steps each way, as decimals and nearest whole numbers.

Why can you add instead of multiply?

Because the golden ratio squared equals itself plus one, so adding two consecutive terms gives the next, the same rule as the Fibonacci sequence.

How is it linked to Fibonacci numbers?

The ratios of consecutive Fibonacci numbers get closer and closer to the golden ratio as the sequence grows.

What is it useful for?

Design scales and proportions, the golden rectangle in art, spirals in nature, and many results in mathematics.

References

On the golden ratio. The golden ratio, about 1.618, is the proportion in which the whole relates to the larger part as the larger relates to the smaller.

1. Eric W. Weisstein, "Golden Ratio," from MathWorld, a Wolfram resource, on the golden ratio, its properties, and its link to the Fibonacci numbers.
2. "Golden ratio," Encyclopaedia Britannica, on the golden ratio as an irrational number about 1.618 and its history from Euclid.