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Calculate perfect golden ratio proportions (φ ≈ 1.618) for design, art, and architecture
Golden Ratio (φ): 1.618033988749895
Also known as phi, the divine proportion, or golden mean
Enter a value to calculate golden ratio proportions
The golden ratio (φ, phi) is a mathematical constant approximately equal to 1.618033988749895. It appears when a line is divided into two parts such that the ratio of the whole length to the larger part equals the ratio of the larger part to the smaller part.
(a + b) / b = b / a = φ ≈ 1.618
b = a × φ
b = a × 1.618034
Where a is the smaller part
a = b ÷ φ
a = b ÷ 1.618034
Where b is the larger part
φ = (1 + √5) / 2
φ ≈ 1.618033988749895
Exact mathematical definition
φ = lim(F(n+1) / F(n))
Ratio of consecutive Fibonacci numbers approaches φ
Example: 89/55 ≈ 1.618182
| Smaller Part (a) | Larger Part (b) | Total (a + b) | Ratio (b/a) |
|---|---|---|---|
| 100.00 | 161.80 | 261.80 | 1.618 |
| 200.00 | 323.61 | 523.61 | 1.618 |
| 300.00 | 485.41 | 785.41 | 1.618 |
| 400.00 | 647.21 | 1047.21 | 1.618 |
| 500.00 | 809.02 | 1309.02 | 1.618 |
| 1000.00 | 1618.03 | 2618.03 | 1.618 |
The golden ratio (φ, phi) is a mathematical constant approximately equal to 1.618. It occurs when a line is divided into two parts so that the ratio of the whole length to the larger part equals the ratio of the larger part to the smaller part. Mathematically: (a+b)/b = b/a = φ. It's also called the golden mean, golden section, or divine proportion.
To calculate the golden ratio: φ = (1 + √5) / 2 ≈ 1.618033988749895. To find golden ratio proportions, multiply the smaller part by 1.618 to get the larger part, or divide the larger part by 1.618 to get the smaller part. For example, if the smaller part is 100, the larger part is 100 × 1.618 = 161.8.
The golden ratio appears throughout nature: in spiral shells (nautilus), flower petals (sunflowers have 55 or 89 petals - Fibonacci numbers), pine cones, seed arrangements, hurricane patterns, galaxy spirals, human DNA, and tree branching patterns. It's also approximated in the proportions of the human body, particularly in the ratio of forearm to hand.
Designers use the golden ratio to create visually pleasing proportions. It's applied in logo design (Apple, Twitter logos), page layouts, typography (font sizes), photography (rule of thirds approximates golden ratio), architecture (building dimensions), and web design (content width ratios). The golden rectangle (width:height = φ:1) is particularly popular.
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...) is deeply connected to the golden ratio. The ratio of consecutive Fibonacci numbers approaches φ as the numbers get larger. For example: 3/2 = 1.5, 5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, 89/55 = 1.618182. As n increases, F(n+1)/F(n) → φ.
To create a golden rectangle, start with a square. Draw a line from the midpoint of one side to an opposite corner. Use that line as a radius to draw an arc that extends the side of the square. The resulting rectangle has proportions of 1:φ (approximately 1:1.618). For example, if the square is 100×100, the golden rectangle will be 100×161.8.