📏 Design Ratio Calculator
Calculate 10 design ratios at once
Input Form
Calculated proportions
| Visual & name | Ratio μₙ | Longer side | Shorter side |
|---|---|---|---|
Plastic number | 1.325 | 132.5 | 75.5 |
Yamato ratio (√2) | 1.414 | 141.4 | 70.7 |
Supergolden ratio | 1.466 | 146.6 | 68.2 |
Golden ratio (1st metallic) | 1.618 | 161.8 | 61.8 |
Platinum ratio (√3) | 1.732 | 173.2 | 57.7 |
Silver ratio (2nd metallic) | 2.414 | 241.4 | 41.4 |
Bronze ratio (3rd metallic) | 3.303 | 330.3 | 30.3 |
Copper ratio (4th metallic) | 4.236 | 423.6 | 23.6 |
Nickel ratio (5th metallic) | 5.193 | 519.3 | 19.3 |
Iron ratio (6th metallic) | 6.162 | 616.2 | 16.2 |
Note
Features
Enter a single base value to instantly see the longer and shorter sides for multiple metallic means including the golden ratio, silver ratio, and bronze ratio. Each ratio value can be copied with one click.
How to Use
Enter your base value and results for six metallic means (golden through iron) will appear immediately. You can adjust decimal places from 0 to 6.
Use Cases
Comparing design layouts, evaluating multiple ratios simultaneously, proportion planning for architecture and interiors, and learning or teaching mathematical ratios.
FAQ
Q: What are metallic means? A: For positive integer n, the metallic mean μₙ is defined as (n + √(n² + 4)) ÷ 2. This creates an infinite family of irrational numbers, where n=1 gives the golden ratio and n=2 gives the silver ratio. They're named after metals: golden, silver, bronze, etc. Q: What's the formula for calculating these ratios? A: The formula is μₙ = (n + √(n² + 4)) / 2. As n increases, the ratio value also increases. When expressed as continued fractions, they take the beautiful form [n; n, n, …]. Q: What is the plastic number? A: An irrational number approximately 1.325, the real solution to x³ = x + 1. Discovered by 20th-century Dutch architect Dom Hans van der Laan, who proposed it as the most pleasing three-dimensional proportion for human perception in architectural design. Q: What is the Yamato ratio? A: The ratio √2 (approximately 1.414), traditionally used in Japanese architecture and art. It's the aspect ratio of A and B series paper sizes (1:√2), with the convenient property that the ratio remains unchanged when folded in half. Q: What is the supergolden ratio? A: An irrational number approximately 1.466, the real solution to x³ = x² + 1. It can be considered a three-dimensional analog of the golden ratio and is applied to proportions in three-dimensional design and structures. Q: What is the golden ratio? A: The ratio approximately 1.618, the most famous "beautiful" proportion. It's the first metallic mean (n=1) and appears in the Parthenon, the Mona Lisa, Apple product designs, and many other artworks and products. It's also the limit of adjacent term ratios in the Fibonacci sequence. Q: What is the platinum ratio? A: The ratio √3 (approximately 1.732), which appears in the relationship between the height and half-base of an equilateral triangle. It's related to regular hexagons and honeycomb structures, forming nature's most efficient packing pattern. Q: What is the silver ratio? A: The ratio approximately 2.414, the second metallic mean (n=2). In Japan, it has been traditionally called the "silver ratio" and appears in structures like Horyuji Temple's five-story pagoda and the observation deck positions of Tokyo Skytree. Q: What is the bronze ratio? A: The ratio approximately 3.303, the third metallic mean (n=3). It provides a more elongated proportion than the golden or silver ratios and is useful for vertical designs and architectural elements. Q: What is the copper ratio? A: The ratio approximately 4.236, the fourth metallic mean (n=4). With its more extreme aspect ratio, it's referenced for proportions of slender structures like towers and chimneys. Q: What is the nickel ratio? A: The ratio approximately 5.193, the fifth metallic mean (n=5). With its very elongated proportion, it's occasionally used for specialized design purposes. Q: What is the iron ratio? A: The ratio approximately 6.162, the sixth metallic mean (n=6). Among the metallic means, it has a particularly elongated ratio and is referenced when expressing extreme proportions.
Tips
Metallic means can be expressed as beautiful continued fractions [n; n, n, …]. The golden ratio is the limit of the Fibonacci sequence (1, 1, 2, 3, 5, 8...) ratios of adjacent terms, appearing in nature in sunflower seed arrangements and nautilus shells. The Yamato ratio (√2) is used for A and B series paper—when folded in half, the aspect ratio remains the same. The platinum ratio (√3) relates to regular hexagons and honeycomb structures, forming nature's most efficient packing pattern. The plastic number was discovered by a 20th-century Dutch architect who proposed it as the most pleasing three-dimensional proportion for human perception.