Golden Ratio Calculator
The golden ratio, represented by the Greek letter phi and approximately equal to 1.6180339887, is one of the most famous numbers in mathematics. It describes the unique way a line segment can be divided into two parts so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part. In other words, if you split a length of 100 into a longer segment A and a shorter segment B, the golden ratio requires that 100 divided by A equals A divided by B, and both ratios equal phi. This self-referential property is the source of its mathematical elegance and is why the ratio appears in the Fibonacci sequence, where the ratio of consecutive terms converges to phi as the sequence grows. In design and art, golden ratio proportions are often considered visually harmonious, and they are used in logo design, photography, typography, and architecture. You can enter any one of three values: the total length, the longer segment, or the shorter segment, and the calculator will derive the other two using phi. It also returns the exact ratio of longer to shorter as a confirmation, and the dimensions of the corresponding golden rectangle where the width is the total and the height is the longer segment. You can use any unit, since phi is a dimensionless ratio. This tool is useful for designers, architects, students exploring mathematical proportions, and anyone who wants to apply the golden ratio to a specific measurement.
Phi = (1 + √5) / 2 ≈ 1.6180339887. Values are displayed to 2 decimal places.
How it works
The golden ratio phi equals (1 plus the square root of 5) divided by 2, approximately 1.6180339887. Given a total length T, the longer segment is T divided by phi (which equals T times (phi minus 1), since 1 divided by phi equals phi minus 1). The shorter segment is T minus the longer segment, which equals T divided by phi squared. If you enter the longer segment A instead, the total is A times phi and the shorter segment is A divided by phi, which is A times (phi minus 1). If you enter the shorter segment B, the longer segment is B times phi and the total is B times phi squared. The ratio of longer to shorter always equals phi when all three conditions are satisfied.
Worked example
Using the default total of 100: longer segment = 100 divided by phi = 100 divided by 1.6180339887 = 61.80 (rounded to 2 decimal places). Shorter segment = 100 minus 61.80 = 38.20. Ratio check: 61.80 divided by 38.20 = 1.618, confirming the golden ratio is achieved. The corresponding golden rectangle would have dimensions 100 by 61.80.
Related calculators
- Fibonacci Calculator: find the nth Fibonacci number; consecutive ratios converge to phi.
- Proportion Calculator: solve a/b = c/d for any missing value.
- Geometric Sequence Calculator: nth term and sum of a geometric progression.
- Percentage Calculator: find what percentage one length is of another.