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Golden Ratio Calculator

Use this free golden ratio calculator to work with phi (φ = 1.6180339887). Enter the long side or the short side of a golden rectangle to find the other side, or split any number into two parts in golden ratio — and see the first eight Fibonacci numbers and how their ratios approach phi.

How it works

The golden ratio is a single number with a very particular property. Take two lengths, a longer one and a shorter one. If the longer divided by the shorter equals the whole divided by the longer, then that ratio is the golden ratio, and it always equals the same number: 1.6180339887…, written with the Greek letter phi (φ). It is the only positive number that is exactly 1 more than its own reciprocal. That self-similarity is why the ratio keeps turning up in geometry, art, and nature.

A golden rectangle is a rectangle whose sides are in golden ratio. If you know the long side, divide it by φ to get the short side. If you know the short side, multiply it by φ to get the long side. The rectangle has a famous trick: cut off the largest square you can from one end, and the rectangle that is left over is itself a golden rectangle, just smaller. Cut a square off that one, and the leftover is golden again. You can repeat the process forever, and the squares spiral inward — which is where the golden spiral comes from, the curve you see drawn through the corners of those squares.

Splitting a number in golden ratio uses the same idea. To divide a total into a larger part and a smaller part so that the larger is to the smaller as the whole is to the larger, multiply the total by 1 ÷ φ (about 0.6180339887) for the larger part, and by 1 ÷ φ² (about 0.3819660113) for the smaller part. The two parts always add up to the total, and their ratio is φ. Designers use this split to place a headline, a photo, or a sidebar at the golden point of a page — about 62% across — because the eye finds that balance pleasing. Splitting 100 in golden ratio gives about 61.80 and 38.20, and 61.80 ÷ 38.20 is 1.618.

The golden ratio is also tied to the Fibonacci sequence, where each number is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. Divide each Fibonacci number by the one before it and the answers wobble above and below φ, getting closer every time: 3 ÷ 2 is 1.5, 5 ÷ 3 is 1.667, 8 ÷ 5 is 1.6, 13 ÷ 8 is 1.625, 21 ÷ 13 is about 1.615, and 34 ÷ 21 is about 1.619. As the numbers grow, the ratio settles onto φ exactly. That is no coincidence: the Fibonacci rule and the golden ratio are two faces of the same mathematics, because φ satisfies the equation φ² = φ + 1, which mirrors the add-the-previous-two rule of the sequence.

Where does the ratio actually appear? In nature, sunflower seeds pack into spirals whose counts are Fibonacci numbers, and the nautilus shell grows in a logarithmic spiral close to the golden one. In art and architecture, the Parthenon and the paintings of the Renaissance are often said to use golden proportions, though some of those claims are exaggerated after the fact. In modern design, the ratio is a practical tool rather than a mystical rule: book covers, web layouts, logos, and photo crops often use golden-rectangle proportions because the balance feels natural. The honest view is that φ is real, beautiful mathematics that shows up often — and that not every rectangle you admire is secretly golden.

To use the calculator, pick what you know: the long side, the short side, or a number to split. Enter the value and the calculator shows both sides of the rectangle, the ratio, and — in split mode — the two parts. The Fibonacci list below the result shows the first eight numbers and how their consecutive ratios close in on φ, so you can watch the connection happen with real numbers.

Worked example

Find the short side of a golden rectangle whose long side is 10.

Step 1: Write down φ. φ = (1 + √5) ÷ 2 = 1.6180339887…
Step 2: The short side is the long side divided by φ: short = 10 ÷ 1.6180339887.
Step 3: Divide: 10 ÷ 1.6180339887 = 6.180339887…
Step 4: Round to two decimal places: the short side is 6.18.
Step 5: Check the ratio: 10 ÷ 6.18 = 1.618, which matches φ to three decimal places. The rectangle is 10 by 6.18.

So a golden rectangle with a long side of 10 has a short side of 6.18. If you instead split 10 in golden ratio, the larger part is 10 ÷ φ = 6.18 and the smaller part is 10 ÷ φ² = 3.82, and 6.18 + 3.82 = 10. Enter 10 in the calculator above, in either the long-side mode or the split mode, and you will get the same numbers.

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