Golden Ratio Calculator — Solve for A, B, or A + B
Calculate the golden ratio φ ≈ 1.618 from one value — the longer segment (A), shorter segment (B), or their sum (A+B) — with the other two and full steps.
What is the golden ratio?
The golden ratio — also called the golden mean or divine proportion, and written φ (the Greek letter phi) — is the number that makes this relationship true for two quantities A (the larger part) and B (the smaller part):
In words: the whole compared to the larger part is the same ratio as the larger part compared to the smaller part. Solving that equation for φ gives a single fixed value:
Because √5 never terminates, φ is irrational — it has no exact fraction or finite decimal form. That makes it the odd one out among this site's math calculators: fractions, ratios, and percents all have exact answers, but φ can only ever be approximated, however many digits you compute.
Using the calculator
Pick which value you already know using the toggle — a (the longer segment), b (the shorter segment), or a + b (their sum) — then type it into the box. The calculator fills in the other two so that all three keep the golden proportion:
- Know a? → b = a / φ, and a+b = a + b.
- Know b? → a = b × φ, and a+b = a + b.
- Know a+b? → a = (a+b) / φ, and b = (a+b) − a.
Every value is shown to 10 significant digits, and the "check" line re-displays a/b so you can confirm it lands on φ.
Worked examples
Example 1 — know a = 10 (the longer segment)
- b = a / φ = 10 / 1.618033989 ≈ 6.180339887.
- a + b = 10 + 6.180339887 ≈ 16.18033989.
- Check: a / b = 10 / 6.180339887 ≈ 1.618033989 = φ. ✓
Example 2 — know b = 5 (the shorter segment)
- a = b × φ = 5 × 1.618033989 ≈ 8.090169944.
- a + b = 8.090169944 + 5 ≈ 13.09016994.
- Check: a / b = 8.090169944 / 5 ≈ 1.618033989 = φ. ✓
Example 3 — know a + b = 100 (splitting a whole)
- a = (a+b) / φ = 100 / 1.618033989 ≈ 61.80339887.
- b = (a+b) − a = 100 − 61.80339887 ≈ 38.19660113.
- Check: a / b = 61.80339887 / 38.19660113 ≈ 1.618033989 = φ. ✓
Golden ratio vs. the Fibonacci sequence
The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… — is built by adding the two previous numbers together, the same self-similar rule that defines the golden ratio (each piece relates to the next the way the next relates to the whole). Divide each Fibonacci number by the one before it, and the result creeps toward φ:
| Fibonacci pair | Ratio | ≈ φ? |
|---|---|---|
| 5 / 3 | 1.6667 | close |
| 13 / 8 | 1.6250 | closer |
| 34 / 21 | 1.6190 | closer still |
| 89 / 55 | 1.6182 | very close |
The further out you go, the closer the ratio gets to 1.6180339887…, but it never lands exactly — Fibonacci numbers are whole (rational), while φ itself is irrational.
Where the golden ratio shows up
Beyond the math, φ shows up as a design heuristic wherever a "balanced but not symmetric" split is wanted — page layouts, logo construction, photo cropping (a subject placed at roughly 62% across the frame), and typographic size scales. It also appears in nature as an emergent pattern (spiral phyllotaxis in sunflower seed heads and pinecones, for instance), because Fibonacci-like growth rules tend to approach it. It is a useful proportion to have in a design toolkit — not a rule that every good composition must follow.
Precision note
φ is computed from √5 using 30-significant-digit decimal arithmetic internally, then rounded only for display (10 significant digits) — so the derived a, b, and a+b values stay consistent with each other however large or small the number you start from.
Frequently asked questions
- What is the golden ratio?
- The golden ratio, written φ (phi), is the number that satisfies (A+B)/A = A/B for two quantities A and B. In other words, the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part. Its value is φ = (1+√5)/2 ≈ 1.6180339887…, an irrational number that never terminates or repeats.
- What is the formula for the golden ratio?
- φ = (1 + √5) / 2. Because √5 is irrational, φ can only be approximated in decimal form — this calculator computes it to 30 significant digits internally and displays 10, so results stay accurate however you round them.
- How is the golden ratio related to the Fibonacci sequence?
- As the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34…) grows, the ratio of each number to the one before it gets closer and closer to φ. For example 34/21 ≈ 1.6190 and 55/34 ≈ 1.6176 — both near 1.618, converging tighter the further along the sequence you go.
- If I know A (the longer segment), how do I find B?
- Divide A by φ: B = A / φ. For example, if A = 10, then B = 10 / 1.618033989 ≈ 6.180339887, and the whole (A+B) ≈ 16.18033989.
- If I know B (the shorter segment), how do I find A?
- Multiply B by φ: A = B × φ. For example, if B = 5, then A = 5 × 1.618033989 ≈ 8.090169944, and A+B ≈ 13.09016994.
- If I only know the total (A+B), how do I split it at the golden ratio?
- Divide the total by φ to get the larger part: A = (A+B) / φ. Then the smaller part is whatever remains: B = (A+B) − A. For a total of 100, A ≈ 61.80339887 and B ≈ 38.19660113.
- Where does the golden ratio show up in design?
- Designers use golden-ratio proportions (roughly 1 : 1.618, or about 62% to 38%) for page layouts, logo grids, photo cropping, and typographic scales, because the split reads as balanced without being a plain 50/50 or 2/3 split. It's a guideline, not a law of aesthetics — plenty of well-loved designs ignore it entirely.
- Why does this calculator need decimal arithmetic instead of exact fractions?
- φ contains √5, an irrational number with no exact finite decimal or fraction representation. Every other value on this site that can be expressed exactly (fractions, ratios, percents) uses exact integer arithmetic; φ is the one case here where a decimal approximation (to 10 significant digits) is the correct and honest answer.