Calculate the Golden Ratio
Calculate the golden ratio of 100: the longer part is 100 ÷ 1.618034 = 61.8034, the shorter is 38.1966, and 100 is to 61.8034 as 61.8034 is to 38.1966.
Golden split
- Longer part 100 ÷ φ, which is 100 × 0.618034.
- 61.8034
- Shorter part 100 - 61.8034, which is also 100 ÷ φ².
- 38.1966
- Check Both ratios are φ: the whole is to the longer part as the longer part is to the shorter.
- 100 ÷ 61.8034 = 1.618 and 61.8034 ÷ 38.1966 = 1.618
- Extended by φ If 100 is the longer part, the whole is 100 × φ = 161.8034 and the shorter part is 61.8034.
- 161.8034
- If it is the shorter part Multiply by φ for the longer part and by φ² = φ + 1 = 2.618034 for the whole.
- longer 161.8034, whole 261.8034
- φ (1 + √5) ÷ 2, an irrational number, so no ratio of whole numbers equals it exactly.
- 1.6180339887...
How the Ratio Calculator works
The golden ratio, φ (phi), is (1 + √5) ÷ 2 = 1.6180339887.... Two lengths are in the golden ratio when the whole is to the longer part as the longer part is to the shorter: (a + b) ÷ a = a ÷ b = φ.
To split a length in the golden ratio, divide it by φ for the longer part and subtract for the shorter. For 100: 100 ÷ 1.618034 = 61.8034, and 100 - 61.8034 = 38.1966. Check: 100 ÷ 61.8034 = 1.618, and 61.8034 ÷ 38.1966 = 1.618.
To extend a length, multiply it by φ. With 100 as the shorter side of a golden rectangle, the longer side is 161.8034, and the two together make 261.8034.
Two identities make the arithmetic quick. 1 ÷ φ = φ - 1 = 0.618034, so dividing by φ is multiplying by 0.618034. And φ² = φ + 1 = 2.618034, so the whole is 2.618 times the shorter part.
φ is irrational, so no ratio of whole numbers equals it, but consecutive Fibonacci numbers come closer and closer: 5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.6154 and 34/21 = 1.6190. A 5 by 8 or 8 by 13 rectangle is the usual whole-number stand-in.
Cutting a square off the end of a golden rectangle leaves a smaller golden rectangle, and repeating the cut forever traces the golden spiral.
Ratio Calculator FAQ
What is the golden ratio of 50?
30.9017 and 19.0983. 50 ÷ 1.618034 = 30.9017 for the longer part, and 50 - 30.9017 = 19.0983 for the shorter.
How do I split a 1920 pixel width in the golden ratio?
1920 ÷ 1.618034 = 1,186.6253, so the columns are about 1,187 and 733 pixels wide.
Is 0.618 the golden ratio too?
It is the same proportion read the other way: 0.618034 is 1 ÷ φ, the shorter part divided by the longer, while 1.618034 is the longer divided by the shorter.