📐 Design Ratio Calculator

01. One base value gives you all ten ratios at once

The six metallic means — golden, silver, bronze and the rest — plus the plastic number, Yamato ratio, supergolden ratio and platinum ratio are all calculated together, so you can compare before committing to one.

02. Both the longer and shorter side are shown, so either can be your starting point

You get the longer side when your input is the short one, and the shorter side when it's the long one, whichever dimension you already have fixed.

03. Copy any figure with a single click

Ratio values, longer sides and shorter sides all copy straight to your clipboard, which spares you mistyping them into your design tool.

04. A diagram beside each ratio shows the proportion at a glance

Every row carries a small figure representing the long and short sides, making differences in shape visible in a way the numbers alone don't convey.

01. Basic Usage

1. Enter the length you want to work from as the base value
2. Longer and shorter sides for ten design ratios (from the plastic number at ~1.325 to the iron ratio at ~6.162) appear immediately
3. Press the copy button next to the value you need

02. Adjusting the Display

1. Use the decimal places control to show anywhere from 0 to 6 digits
2. Each row carries a small figure showing the split between the longer and shorter side, so you can see the difference at a glance

Q. Are all ten ratios metallic means?

A. No. Six are metallic means: golden, silver, bronze, copper, nickel, and iron. The plastic number, Yamato ratio, supergolden ratio, and platinum ratio have different definitions and are included because they are widely used in design.

Q. What input range is accepted?

A. From 0.001 to 1,000,000. Values outside this range produce no result. No unit is assumed, so you can work in mm, px, cm, or anything else.

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 is μₙ?

A. It's the mathematical symbol for metallic means. μ (mu) is a Greek letter, and n represents a positive integer (1, 2, 3...). μ₁ is the golden ratio, μ₂ is the silver ratio, and so on.

Q. What's the formula for calculating these ratios?

A. For metallic means, μₙ = (n + √(n² + 4)) / 2. As n increases, the ratio value also increases, and as continued fractions they take the form [n; n, n, …]. The Yamato ratio (√2) and platinum ratio (√3) are square roots rather than metallic means, while the plastic number and supergolden ratio are solutions to cubic equations.

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. It is distinct from the Yamato ratio (√2).

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.

01. The Shared Form of Metallic Means

Metallic means can be expressed as beautiful continued fractions [n; n, n, …]. Put 1 in for n and you get the golden ratio, 2 gives the silver ratio, 3 the bronze — each emerging in turn from the same expression. The formula is (n + √(n²+4)) ÷ 2. Larger values of n produce larger ratios; this tool carries the six cases from n = 1 through n = 6.

02. The Golden Ratio and Fibonacci

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 other metallic means have the same relationship. The silver ratio is the limit of adjacent terms in the Pell sequence (1, 2, 5, 12, 29...), where each term is twice the previous plus the one before it. The multiplier in that rule is precisely the n of the metallic mean.

03. The Yamato Ratio and Paper Sizes

The Yamato ratio (√2) is used for A and B series paper—when folded in half, the aspect ratio remains the same. A0 is defined to have an area of exactly one square meter, and each fold in half advances the number to A1, A2, and so on. The 141% setting for enlarging A4 to A3 on a copier comes straight from √2 ≈ 1.414. In Japan the proportion also appears in buildings such as Hōryū-ji, which is why it carries the name Yamato ratio.

04. The Platinum Ratio and Hexagons

The platinum ratio (√3) relates to regular hexagons and honeycomb structures, forming nature's most efficient packing pattern. In a regular hexagon, the distance between opposite sides stands in a ratio of √3 to the length of one side. Since the hexagon encloses a given area with the shortest possible perimeter, bees can build their combs while economizing on wax.

05. The Plastic Number

The plastic number was discovered by a 20th-century Dutch architect who proposed it as the most pleasing three-dimensional proportion for human perception. Its value is about 1.3247, the solution to x³ = x + 1 — a counterpart to the golden ratio being the solution to x² = x + 1. The word 'plastic' here has nothing to do with the material; it derives from the Greek for 'able to be shaped.'

06. The Supergolden Ratio

The supergolden ratio is roughly 1.4656, the solution to x³ = x² + 1. The equation resembles the plastic number's, but the solution is a different value. It is also the limit of adjacent terms in Narayana's cows sequence. The fourteenth-century Indian mathematician Narayana posed it as a problem about a cow bearing one calf a year, with calves beginning to breed in their third year — the same structure as Fibonacci's rabbits.

07. Naming Conventions Vary

This tool treats √2 as the Yamato ratio and 2.414 as the silver ratio, listing them separately. That distinction deserves a word of caution. In Japanese design writing, hakuginhi (silver ratio) most often means √2 ≈ 1.414, so readers used to that convention will find this listing at odds with it. In the mathematical series of metallic means, however, the second metallic mean, 2.414, is the silver ratio. The two are not unrelated: 2.414 − 1 = 1.414, a difference of exactly one. They express the same proportional relationship, measured across differently chosen rectangles. To avoid the confusion, this tool gives the √2 case the separate name Yamato ratio.

08. The Four That Are Not Metallic Means

Of the ten proportions listed here, six are computed from the metallic mean formula: the golden, silver, bronze, copper, nickel, and iron ratios. The remaining four — the Yamato ratio (√2), the platinum ratio (√3), the plastic number, and the supergolden ratio — do not belong to that series. They are included as proportions genuinely used in design, and their values are carried as constants.

09. Larger Ratios Mean Longer Shapes

The table runs from the smallest ratio at the top to the largest at the bottom. The plastic number at about 1.32 sits closest to a square, while the iron ratio at about 6.16 is the most elongated. The small band at the left of each row shows how a rectangle divides at that proportion. It lets you compare elongation by eye, which is hard to judge from the numbers alone.

10. Choosing Decimal Places

You can select from 0 to 6 decimal places. For pixel-based layouts, 0 to 1 works well; for print dimensions, 2 to 3 is a comfortable range. The ratio column alone always displays at three decimal places regardless of your selection. Those are constants rather than calculated results, so varying their precision serves little purpose.