The golden proportion, also called the golden ratio, is the number φ (phi) = 1.6180339887… It is the ratio where a/b = (a + b)/a — the whole relates to the longer part exactly as the longer part relates to the shorter part. Use the calculator to find the golden partner of any length, and learn where the golden ratio appears in art, nature and design.
Golden Proportion Calculator
φ = 1.6180339887… — find the golden partner of any length.
What Is the Golden Ratio?
The golden ratio φ is (1 + √5) ÷ 2 = 1.6180339887… It appears when you divide a line so that the whole line relates to the longer part exactly as the longer part relates to the shorter part. In a golden rectangle, the ratio of length to width is φ. The golden ratio is also the limit of the ratios of consecutive Fibonacci numbers: 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615 — they close in on φ.
The Golden Proportion Formula
If a is the longer part and b is the shorter part, the golden proportion is a/b = (a + b)/a = φ. So a = b × φ and b = a ÷ φ. For example, if the shorter part is 10, the longer part is 10 × 1.618 = 16.18, and the whole is 26.18. Check: 26.18 ÷ 16.18 ≈ 1.618 ✓.
Golden Ratio in Art, Nature and Design
Designers use the golden ratio to lay out pages, logos and rooms that feel balanced — a rectangle 16.18 cm by 10 cm is a golden rectangle. In nature, sunflower spirals, pinecones and nautilus shells follow Fibonacci counts that approach φ. In the human face and body, many measurements sit close to golden proportions. The ratio is a guide for pleasing proportions, not a magic rule — good design still depends on context, contrast and purpose.
Worked Examples
Example 1: A poster is 20 cm wide. For a golden rectangle, the length is 20 × 1.618 = 32.36 cm. Example 2: A golden rectangle is 40 cm long. Its width is 40 ÷ 1.618 = 24.72 cm. Example 3: Split 100 into golden parts: the longer part is 100 × 0.618 = 61.8 and the shorter part is 38.2, because 61.8 ÷ 38.2 ≈ 1.618.
Golden Proportion FAQs
What is the golden ratio in simple terms?
It is the number 1.618…, the ratio where the whole is to the longer part as the longer part is to the shorter part. A rectangle with sides in this ratio is a golden rectangle.
How do you calculate the golden ratio?
φ = (1 + √5) ÷ 2 ≈ 1.618. To find the golden partner of a length, multiply it by φ for the longer part, or divide by φ for the shorter part.
Is the golden ratio the same as the Fibonacci sequence?
They are closely linked: the ratios of consecutive Fibonacci numbers (3/2, 5/3, 8/5, 13/8…) get closer and closer to φ as the numbers grow.
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