Golden Ratio Calculator
Enter one value below to instantly find its golden ratio companion.
Blue square + red rectangle = the whole. The red rectangle has the same proportions as the whole.
What Is the Golden Ratio?
φ = (1 + √5) / 2 ≈ 1.618033988749895. Two numbers a and b (with a > b) are in the golden ratio when a / b = (a + b) / a = φ.
Worked example
Find the larger golden-ratio companion of 10.
10 × 1.618... ≈ 16.18
10 and 16.18 are in the golden ratio.
A second example
Find the smaller golden-ratio companion of 26.
26 ÷ 1.618... ≈ 16.07
26 and 16.07 are in the golden ratio.
Keep going
- The golden ratio is really a fixed scale factor, about 1.618; the scale factor calculator applies any ratio you choose to a length, not just phi. Scale Factor Calculator
Frequently Asked Questions
What is the golden ratio?
The golden ratio, φ (phi), is approximately 1.618. Two quantities are in the golden ratio when the ratio of the larger to the smaller equals the ratio of their sum to the larger.
Where does the golden ratio show up?
It appears in certain spiral patterns in nature (like nautilus shells and sunflower seed heads), and it's been used deliberately in art, architecture, and design for its visually pleasing proportions.
How do you find the larger value in a golden ratio pair?
Multiply the smaller value by φ (≈1.618). To go the other direction, divide the larger value by φ.
What algebraic property makes phi special?
φ is the positive solution to x² − x − 1 = 0, which means φ² = φ + 1 — squaring it gives the same result as just adding 1. Check it: 1.618033988...² ≈ 2.618033988..., and 1.618033988... + 1 ≈ 2.618033988.... That self-referential identity is why φ shows up in continued fractions and recursive patterns.
How does the golden ratio relate to the Fibonacci sequence?
Dividing each Fibonacci number by the one before it (1, 1, 2, 3, 5, 8, 13, 21, 34...) produces a ratio that gets closer and closer to φ as the sequence grows — 21/13 ≈ 1.615 and 34/21 ≈ 1.619, converging toward 1.618033988749895.
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