Golden Ratio Calculator

Longer ÷ shorter = phi Whole ÷ longer = phi Shorter = longer ÷ phi Longer = shorter × phi Whole = longer + shorter

Which part you know

A choice of three: the longer part, the shorter part, or the whole line. It decides which of the fields is your input and which two readings are worked out from it. The whole line is the default, because a length you can measure directly is the usual way into the problem.

The length you know

How long that part is: 100, or 1920, or 0.5. It has to be greater than zero, because zero has no golden section — both parts would be zero — and a negative length is not a length. It is not restricted to whole numbers, and it carries no unit: pixels, millimetres and inches all work the same way.

Longer part

The bigger piece of the split, and the primary reading. Knowing the whole, it is the whole divided by phi; knowing the shorter part, it is that part multiplied by phi; knowing itself, it is the number you typed.

Shorter part

The smaller piece. Knowing the whole, it is what is left when the longer part is taken away; knowing the longer part, it is that divided by phi; knowing itself, it is the number you typed.

Whole line

The two pieces added together, which is the length you started with whenever that was the field you filled. It is the reading that ties the other two together: longer plus shorter always returns it, and that sum is the check that the three lines agree.

Rounding

All three readings are printed to four decimal places, because phi itself never terminates. The three are worked out from the same pair of numbers, so when the length you enter lies on the four-decimal grid, the three printed lines still add up exactly: 61.8034 + 38.1966 is 100.

The first use is a layout: a page, a canvas, a photograph or a frame that should be divided at the golden section, where the whole width is known and the two parts are wanted. The second is the other direction, which is just as common — you have already drawn the long piece and now need the short one, or you have a fixed short piece and the long one has to be derived from it. The third is checking a split somebody else produced: enter the longer part and see whether the whole it implies is the one on the drawing. The fourth is a proportion across scales, where the same ratio is applied to a large and a small version of the same design; because the ratio is dimensionless, nothing needs converting between them. The fifth is the one people arrive with after meeting the phrase rather than the number: seeing that a line of 100 becomes 61.8 and 38.2 is what makes the definition concrete, and the table below repeats that for a handful of familiar widths so the effect of scaling can be read off in one column.

1. A whole line of 100

   1. Longer: 100 ÷ 1.6180339887 = 61.8034
   2. Shorter: 100 − 61.8034 = 38.1966
   3. Check: 61.8034 + 38.1966 = 100, the length you started from
   4. And the ratio: 61.8034 ÷ 38.1966 = 1.618

   The default case and the clearest one: a round 100 in, two awkward numbers out, and the two of them adding back to exactly 100. That last part is not luck — the shorter piece is found by subtraction rather than by a second division, so the two printed lines cannot disagree with the line you entered. The fourth step is the definition run backwards, and it is the step worth doing once: dividing the longer piece by the shorter has to return phi.

2. Known longer part of 100

   1. Shorter: 100 ÷ 1.6180339887 = 61.8034
   2. Whole: 100 + 61.8034 = 161.8034
   3. So a long piece of 100 belongs to a line of 161.8034

   The direction people actually need when the design already has a long edge: the long piece is fixed and the rest of the line has to follow it. Note that the primary reading here is simply the number that was typed — the panel keeps the longer part in the first line whichever branch you took, so the answer is always in the same place. And note the second line is the same 61.8034 as in the case above, which is the whole point: a long piece of 100 implies the identical short piece, whether it came from a line of 161.8 or from a line of 100 being split differently.

3. Known shorter part of 100

   1. Longer: 100 × 1.6180339887 = 161.8034
   2. Whole: 100 + 161.8034 = 261.8034
   3. So a short piece of 100 belongs to a line of 261.8034

   The third branch, and the one that pins down the multiplication: going from the short piece to the long one multiplies by phi, while going from the long piece to the short one divides by it. Both directions appear in the cases above, and comparing them is the quickest way to remember which way round the constant goes. A short piece of 100 needs a line more than two and a half times its length, which is the practical warning: the short piece of a golden section is always the smaller share of a line that is longer than it looks.

4. A 1920 pixel width

   1. Longer: 1920 ÷ 1.6180339887 = 1186.6253
   2. Shorter: 1920 − 1186.6253 = 733.3747
   3. Check: 1186.6253 + 733.3747 = 1920

   A screen width, which is where this calculation is most often performed in practice: a canvas or a content column divided at the golden section. Both parts land on four decimal places, and in pixels that means a fraction of a pixel — round them to whole pixels and the split is still golden to well within what an eye can see. It is also the case that shows why the readings are printed to four decimals rather than to two: at this size, two decimals would put both parts on whole-ish numbers and hide the fact that neither of them is exact.

5. A whole line of 2

   1. Longer: 2 ÷ 1.6180339887 = 1.2361
   2. Shorter: 2 − 1.2361 = 0.7639
   3. Check: 1.2361 + 0.7639 = 2

   The smallest whole-number length that still shows a split with digits on both sides of the decimal point, and the case that makes the ratio's oddity visible: the longer part of a line of 2 is more than half of it, as it always is, and the shorter part is a little under four tenths. Scaling this line by ten gives the 20 that the table below contains, and both parts scale by ten with it — there is no unit and no scale in the ratio, only a proportion between two pieces of whatever you measured.

The length you enter has to be greater than zero. Zero is refused because its golden section would be two zeros, which says nothing, and a negative length is refused because it is a length that does not exist rather than a small one. Only one length is taken at a time, and the page never checks a split you performed elsewhere: if you have two pieces already cut, entering one of them returns the section that piece belongs to rather than confirming the pair you have. The three readings are printed to four decimal places, so the numbers are never the exact ones — phi has no exact decimal form at all, and the page is rounding at every line; when both the whole and the longer part are entered on the grid the printed lines still sum exactly, but a length with five or more decimals can leave the last digit one out, and that is the printing rather than the arithmetic. Lengths are unitless on purpose, which means the page cannot tell you whether the two pieces are millimetres or miles, and it also means it will not warn you that a split in pixels will need rounding to whole pixels to be buildable. The page applies the ratio and nothing else: it does not tell you whether a golden section is the right proportion for the job, and it does not produce the logarithmic spiral that successive golden rectangles are usually drawn with.

What is the golden ratio, exactly?

One plus the square root of five, divided by two, which is 1.6180339887498948482 and goes on forever without repeating. It is the only ratio that survives being cut up: take a rectangle with those sides, remove the largest square, and the rectangle left over has the same ratio again. Euclid described it as dividing a line in extreme and mean ratio, which is the operation this page performs.

How do I split a length in the golden ratio?

Divide the whole by 1.6180339887 to get the longer piece, then subtract that from the whole to get the shorter one. A line of 100 becomes 61.8034 and 38.1966. The subtraction rather than a second division is deliberate: it guarantees that the two pieces add back to the length you entered.

I already know one of the pieces, not the whole. Can I use it?

Yes — choose which part you know in the first field and enter its length. Knowing the longer piece, the shorter is that divided by phi and the whole is the two added. Knowing the shorter piece, the longer is that multiplied by phi. All three branches are the same arithmetic rearranged, and the panel always prints the same three lines.

Why do the numbers come out with four decimal places?

Because phi never terminates, so no reading on this page can be exact — the four places are where the rounding is stopped, in the same place for all three lines. When the length you enter has at most four decimals, the three printed lines still add up exactly, because the shorter piece is found by subtraction. A length with five or more decimals can leave the last digit one out.

Does it matter what unit I use?

No, and that is why no unit appears anywhere on the page. The golden ratio is a proportion between two lengths rather than a length, so pixels, millimetres, inches and plain fractions all behave identically, and scaling the length you enter scales all three readings by the same factor. The consequence is that the page cannot tell you whether your answer needs rounding to whole pixels before it can be built.

Is 1.618 good enough?

For cutting a piece of card, yes; for the definition, no. The golden ratio is not 1.618 — it is one plus the square root of five over two, which only begins with those digits. Using the four-digit 1.618 throughout shifts the longer piece of a 100 line by about a thousandth, which is invisible in most work but is not the golden section. This page uses the full precision available to it and rounds only what it prints.

Isn't this the same as a ratio or a proportion calculator?

No, and the difference is what you have to supply. A ratio calculator takes two numbers you give it and reduces them, and a proportion calculator solves for a missing term in a statement you wrote. This page's ratio is not yours to choose: it is fixed at phi, and what you enter is a length. Three pages, three different questions.

• Golden Ratio — the definition of phi as the unique ratio which survives cutting a rectangle into a square and a remainder, its exact value of one plus the square root of five over two, its other names (divine proportion, golden mean, golden section), and Euclid's equivalent definition by extreme and mean ratio — Wolfram MathWorld (United States)
• Golden Ratio Conjugate — the smaller quantity, equal to phi minus one and to one over phi, which is the ratio of the short piece to the long piece on this page — Wolfram MathWorld (United States)
• Golden Rectangle — the rectangle whose sides are in the golden ratio and which keeps that ratio when a square is cut from it, the figure that makes the definition of phi unique — Wolfram MathWorld (United States)
• Fibonacci Number — the sequence whose terms are connected to the golden ratio, and whose neighbouring pairs approach the same proportion this page applies — Wolfram MathWorld (United States)
• Ratio — the relation between two quantities of the same kind, written as a quotient, which is the form both of the equalities in the expression above take — Wolfram MathWorld (United States)