About the Golden Ratio Calculator
This golden ratio calculator works out the golden section for any length. Enter a total length and it splits it into a longer part (a) and a shorter part (b) so that the whole is to the longer part as the longer part is to the shorter — the proportion φ ≈ 1.6180339887. Or enter just the longer or the shorter segment and it finds the other two.
Designers, photographers, architects and woodworkers use it to size layouts, crop images, place focal points, set column widths or proportion furniture; students use it to check φ problems. Any unit works — pixels, inches, centimetres or feet — because the golden ratio is a pure proportion.
The results also show the golden rectangle: a rectangle whose long side is the longer segment and short side the shorter segment, which keeps the same shape when a square is cut off it.
With the default inputs, the longer segment (a) is 61.8034. Change any value above to recalculate instantly.
How to use the golden ratio calculator
- 1Choose which length you already know: the total, the longer part or the shorter part.
- 2Enter the length in any unit.
- 3Read the longer segment, shorter segment and total from the results.
- 4Use the two segments as the sides of a golden rectangle or as a layout split.
Formula and method
Two lengths a > b are in the golden ratio when the ratio of their sum to the larger equals the ratio of the larger to the smaller. Solving (a + b)/a = a/b gives φ² = φ + 1, whose positive root is φ = (1 + √5)/2 ≈ 1.6180339887.
So the longer segment is the total divided by φ (about 61.8% of it), and the shorter segment is the longer divided by φ (about 38.2% of the total). Given the shorter segment, the longer is b × φ; given the longer, the total is a × φ. Because 1/φ = φ − 1, the two shares always add up to exactly 100%.
- φ
- Golden ratio ≈ 1.6180339887
- a
- Longer segment
- b
- Shorter segment
Worked examples
Split 100 units into a golden section
The longer part is 100 ÷ 1.618034 ≈ 61.80 and the shorter part is 61.80 ÷ 1.618034 ≈ 38.20. Their sum is 100, and 100/61.80 = 61.80/38.20 = φ.
Golden rectangle from a 10-inch side
If 10 in is the longer segment, the shorter is 10 ÷ φ ≈ 6.18 in, and the whole length is 10 × φ ≈ 16.18 in. A 10 × 6.18 in rectangle is a golden rectangle.
Find the long side from a 5 cm short side
Multiplying the short side by φ gives the long side: 5 × 1.618034 ≈ 8.09 cm. The total is 5 + 8.09 ≈ 13.09 cm.
Frequently asked questions
What is the golden ratio?+
The golden ratio, φ (phi), is about 1.6180339887. Two quantities are in the golden ratio when the ratio of their sum to the larger one equals the ratio of the larger to the smaller. It is an irrational number equal to (1 + √5) ÷ 2.
How do I divide a line by the golden ratio?+
Divide the total length by 1.618 to get the longer part, then subtract it from the total (or divide it by 1.618 again) to get the shorter part. The split is about 61.8% and 38.2%.
What is a golden rectangle?+
A golden rectangle has sides in the ratio 1 : 1.618. If you cut a square off one end, the remaining rectangle is again a golden rectangle, which is how the golden spiral is drawn.
How is the golden ratio related to Fibonacci numbers?+
The ratio of consecutive Fibonacci numbers — 8/5, 13/8, 21/13, 34/21 — gets closer and closer to φ. That is why Fibonacci numbers are often used as easy approximations of golden proportions.
Is the golden ratio the same as the rule of thirds?+
No. The rule of thirds splits a frame at 33.3% and 66.7%, while the golden section splits it at about 38.2% and 61.8%. The golden grid (phi grid) places lines slightly closer to the centre than the rule of thirds.