Golden Rectangle Calculator

What this calculator does

A golden rectangle is a rectangle whose two sides are in the golden ratio, roughly 1.618 to 1, a shape often called the most pleasing proportion in design. This builds one for you from a single side, and shows the smaller golden rectangle nested inside it.

Enter a side length and it returns the rectangle's dimensions. It runs right here in the browser.

Using the calculator

1. Enter a side length.
2. Press Calculate.

It returns the outer golden rectangle, your side by that side times the golden ratio, and the interior golden rectangle left when a square is removed. Reset clears the box.

What a golden rectangle is

Take a rectangle and look at the ratio of its long side to its short side. For most rectangles that ratio is nothing special. But when it equals the golden ratio, about 1.618, you have a golden rectangle. So if the short side is 1, the long side is roughly 1.618.

The golden ratio itself, written with the Greek letter phi, has the exact value (1 + √5) ÷ 2, which works out to 1.6180339 and on forever. This tool uses the familiar rounded value of 1.618, which is more than close enough for any practical drawing or layout.

The square trick, and why it never ends

Here is the property that makes the golden rectangle remarkable, and it is exactly what the tool shows you. Take a golden rectangle and slice off the largest square you can from one end. What remains is a smaller rectangle, and that smaller rectangle is also golden, with its sides in the very same 1.618 ratio.

Which means you can do it again. Cut a square off the smaller golden rectangle and you get an even smaller golden rectangle, and so on, forever, each a perfect scaled-down copy of the last. This self-similarity, a shape that contains endless smaller versions of itself, is unique to the golden ratio, and it is why the golden rectangle turns up wherever nested, self-repeating proportion is wanted.

The golden spiral

Those nested squares do something beautiful. If you draw a quarter-circle across each square, corner to corner, and let them flow one into the next, the arcs join into a smooth curve that winds inward forever: the golden spiral. It is the elegant logarithmic spiral you see drawn over nautilus shells, spiralling galaxies and sunflower heads, and it grows outward by the golden ratio with every quarter turn.

A worked example

Start with a side of 10.

1. The long side is 10 × 1.618 = 16.18, so the golden rectangle is 16.18 by 10.
2. Remove a 10 by 10 square, and the strip left over is 10 by 6.18 (that is 16.18 − 10).
3. Check it is golden: 10 ÷ 6.18 ≈ 1.618. It is, so the interior rectangle is a smaller golden rectangle, just as expected.

In art and design

The golden rectangle has long been associated with beauty and balance, and you will find it credited in everything from the Parthenon and Renaissance paintings to modern company logos. It is only fair to add that how deliberately the ratio was used in many older works is genuinely debated by historians, and some famous claims are more legend than fact. What is not in doubt is that designers today reach for it on purpose, using it to size layouts, frames and compositions, because proportions near the golden ratio tend to feel comfortably balanced to the eye.

Questions people ask

What is a golden rectangle?

A rectangle whose long side divided by its short side equals the golden ratio, about 1.618.

What is the golden ratio?

The number phi, exactly (1 + √5) ÷ 2, which is about 1.618. It is the ratio of the sides of a golden rectangle.

What happens if I remove a square from it?

You are left with a smaller rectangle that is itself golden, with the same 1.618 ratio. You can repeat this endlessly, each time getting a smaller golden rectangle.

What is the golden spiral?

The curve formed by drawing quarter-circles through the nested squares of a golden rectangle. It spirals inward forever and appears in shells and galaxies.

Is it really used in art and architecture?

It is widely associated with both, and designers use it deliberately today. Many historical claims, though, are debated, so it is best treated as a well-loved guide to proportion rather than a proven rule.

References

A note on where this comes from. A golden rectangle has its sides in the golden ratio, phi, equal to (1 + √5) ÷ 2, the proportion Euclid described as division in extreme and mean ratio around 300 BCE and Luca Pacioli later called the divine proportion. For further reading, see Golden rectangle.

1. The golden ratio, phi = (1 + √5) ÷ 2 ≈ 1.618, the ratio of the sides of a golden rectangle.
2. Euclid, Elements (c. 300 BCE), which defines the division of a line in extreme and mean ratio, the golden ratio.