Golden Section Calculator

What this calculator does

The golden section is the special way to cut a line into two pieces so that they relate in the golden ratio: the whole line is to the larger piece exactly as the larger piece is to the smaller. This finds that cut, working from whichever part you already know.

Enter the whole, the larger part, or the smaller part, and it fills in the rest. It runs right here in the browser.

Using the calculator

1. Enter any one of the three: the whole length (a + b), the larger part (a), or the smaller part (b).
2. Press Calculate.

It works out the other two so that the three sit in the golden proportion. Reset clears the fields.

What the golden section is

Imagine a line split into a longer part, call it a, and a shorter part, b. Almost any split gives two unrelated lengths. But there is one special place to cut where a lovely balance holds: the whole line compared to the longer part is in exactly the same ratio as the longer part compared to the shorter. That balance point is the golden section, and the ratio it produces is the golden ratio, about 1.618.

Euclid wrote about this cut around 300 BCE, calling it dividing a line "in extreme and mean ratio". It is the oldest known description of the golden ratio, long before the shape had its modern name.

The proportion, and the two magic points

Written out, the defining relationship is:

(a + b) ÷ a = a ÷ b = 1.618

Working that through gives two tidy fractions of the whole. The longer part comes to about 0.618 of the whole line, and the shorter part to about 0.382. Those two decimals add up to 1, as they must, since together they are the whole line. They are the golden section points, and if you have ever heard that 0.618 is a "magic" proportion, this is where it comes from.

Solving from any part

Because the three lengths are locked together by that single proportion, knowing any one of them fixes the other two. Give the tool the whole and it splits it at roughly 0.618 and 0.382. Give it the longer part and it finds the shorter part (the longer divided by 1.618) and adds them for the whole. Give it the shorter part and it scales up to the longer (the shorter times 1.618) and the whole. Whichever piece you have, the rest follows.

A worked example

Take a whole line of 100 and find its golden section.

1. The longer part is about 100 × 0.618 = 61.8.
2. The shorter part is the remainder, 100 − 61.8 = 38.2.
3. Check the proportion: 100 ÷ 61.8 ≈ 1.618, and 61.8 ÷ 38.2 ≈ 1.618. The ratio is the same at both scales, which is the whole point of the golden section.

Where it turns up

The golden section is a favourite tool of designers and artists. Placing a focal point at the golden section of a frame, rather than dead centre, often gives a composition a natural, comfortable balance, and photographers use a close cousin of this idea in the rule of thirds. It is also the same proportion, applied to a rectangle's sides, that produces the golden rectangle, and the same number the Fibonacci sequence homes in on. If you want the ratio itself rather than a divided line, the golden ratio calculator handles that.

Questions people ask

What is the golden section?

A division of a line into two parts so that the whole is to the larger part as the larger is to the smaller, a ratio equal to the golden ratio, about 1.618.

What did Euclid mean by extreme and mean ratio?

It is his name for this same cut. The "mean" is the larger part, sitting between the whole and the smaller part in equal ratio, which is exactly the golden section.

What are the 0.618 and 0.382 points?

They are the fractions of the whole line taken by the larger and smaller parts. Together they make 1, since they add up to the whole line.

How do I divide a line in the golden ratio?

Multiply the length by 0.618 for the longer part; the rest is the shorter part. This tool does it, and works backwards from either part too.

Where is it used?

In art and design for balanced composition, in a related form as the rule of thirds in photography, and as the proportion behind the golden rectangle.

References

A note on where this comes from. The golden section divides a line so that the whole is to the larger part as the larger is to the smaller, giving the golden ratio phi. Euclid set it out in his Elements around 300 BCE as division in extreme and mean ratio, the earliest known treatment of the golden ratio. For further reading, see Golden ratio.

1. Euclid, Elements (c. 300 BCE), defining the division of a line in extreme and mean ratio.
2. The golden ratio, phi = (1 + √5) ÷ 2 ≈ 1.618, the ratio produced by the golden section.