The golden ratio, written with the Greek letter φ (phi), is the proportion you get when a length is cut so that the whole is to the longer piece as the longer piece is to the shorter one. Its value is about 1.618. Designers use it to choose pleasing proportions for layouts, logos and frames, and it turns up in the geometry of pentagons and the growth patterns of some plants. This calculator takes any one length — the longer part, the shorter part or the whole — and finds the other two.
How to use the golden ratio calculator
- Choose which length you know: the longer part (a), the shorter part (b) or the whole (a + b).
- Enter the length and, optionally, a unit such as
in,cmorpx. - The tape lists all three lengths and the matching golden rectangle. The drawing shows the rectangle split into a square and a smaller golden rectangle, with the first arcs of the golden spiral.
Golden ratio formulas
The defining proportion is:
Setting b = 1 gives φ² = φ + 1, a quadratic equation whose positive solution is:
From that, every conversion is a single multiplication or division:
| You know | Longer part a | Shorter part b | Whole a + b |
|---|---|---|---|
| a | — | a ÷ φ ≈ 0.618 a | a × φ |
| b | b × φ | — | b × φ² ≈ 2.618 b |
| a + b | (a + b) ÷ φ | (a + b) ÷ φ² | — |
A useful curiosity: 1/φ = φ − 1 ≈ 0.618, so dividing by φ is the same as multiplying by 0.618.
Worked example
A designer wants a card whose short side is 10 inches and whose proportions are golden.
Known: a = 10 in (the short side of the card becomes the longer golden part)
Shorter part: b = 10 ÷ 1.618034 ≈ 6.18 in
Long side of the card: a + b = 10 × 1.618034 ≈ 16.18 in
Check: 16.18 ÷ 10 ≈ 1.618 and 10 ÷ 6.18 ≈ 1.618
Cutting a 10 × 10 square from the card leaves a 6.18 × 10 rectangle — itself golden. Repeat the cut and the squares spiral inward forever, which is how the golden spiral in the drawing is built.
The Fibonacci connection
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 — each number is the sum of the two before it. Dividing each term by the one before it homes in on φ:
| Ratio | Value |
|---|---|
| 3/2 | 1.5 |
| 5/3 | 1.6667 |
| 8/5 | 1.6 |
| 13/8 | 1.625 |
| 21/13 | 1.6154 |
| 34/21 | 1.6190 |
| 89/55 | 1.6182 |
That is why designers often use Fibonacci pairs such as 5 : 8 or 8 : 13 as practical stand-ins for the golden ratio. Explore the sequence itself with the Fibonacci calculator.
Where the golden ratio appears — and where it doesn’t
- Pentagons and stars: in a regular pentagon, the ratio of a diagonal to a side is exactly φ. A five-pointed star is full of golden ratios.
- Plant growth: many sunflowers and pinecones place seeds at the “golden angle,” about 137.5°, which is 360° divided by φ². The resulting spirals usually come in Fibonacci counts.
- Design: grid systems, typography scales and photo crops sometimes use φ as a guide, though the rule of thirds and simple ratios like 3 : 2 are equally common.
- Exaggerated claims: measurements of the Parthenon, the Great Pyramid or the Mona Lisa that “prove” the golden ratio usually depend on where you choose to measure. Treat them with healthy skepticism.
For general proportions that are not golden, use the ratio calculator. To size a rectangle’s area and perimeter once you know its sides, the rectangle calculator completes the job.
Frequently asked questions
What is the golden ratio?
It is the number φ (phi) = (1 + √5) ÷ 2 ≈ 1.6180339887. Two lengths a > b are in the golden ratio when the whole a + b relates to a exactly as a relates to b.
How do I divide a line in the golden ratio?
Divide the total length by φ to get the longer part, then subtract to get the shorter part. A 100 cm line splits into about 61.80 cm and 38.20 cm.
What are the sides of a golden rectangle?
The long side is φ times the short side. A golden rectangle that is 10 inches tall is about 16.18 inches wide. Cutting a 10 × 10 square off one end leaves a smaller rectangle, about 6.18 × 10, that is also golden.
How is the golden ratio related to Fibonacci numbers?
The ratio of consecutive Fibonacci numbers approaches φ. 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.6154 and 89/55 ≈ 1.6182, getting closer with every step.
Is the golden ratio really everywhere in art and nature?
Less than popular claims suggest. It does appear in the geometry of regular pentagons and in the spiral arrangement of seeds in sunflowers, but many claims about famous buildings and paintings rely on loosely chosen measurements.