Golden Rectangle Calculator

Background.

Most people who want a golden rectangle already have one dimension — the short edge of a card, the width of a wall, the area of canvas they can afford — and need the rest. For the sides alone it is quick: multiply the short side by φ, about 1.618, or divide the long side by it. Working back from an area, a perimeter or a diagonal takes a square root or two, and those routes are here as well.

The ratio comes from Euclid's "extreme and mean ratio" in Book VI of the Elements: a line cut so that the whole is to the larger part as the larger part is to the smaller. For a rectangle with long side a and short side b that reads (a + b)/a = a/b. Call a/b φ and it becomes φ² = φ + 1, whose positive root is (1 + √5)/2 = 1.6180…, an irrational number whose decimals never end or repeat. A golden rectangle therefore never has two whole-number sides; one of them is always rounded.

What makes the shape special is self-similarity. Cut a square off the short side and the piece left over is again a golden rectangle, smaller by a factor of φ. Do it again to the leftover and the process never ends; quarter-circles drawn through the squares trace the spiral that usually illustrates the ratio. The leftover's short side is reported so you can start that subdivision.

Claims about where the ratio occurs deserve more doubt than the arithmetic. George Markowsky's 1992 survey in the College Mathematics Journal examined the standard examples — the Parthenon, the Great Pyramid, the Mona Lisa, the proportions of the human body — and found that they do not stand up. Pick the proportion because you like how it looks, not because history supposedly endorses it.

What is golden rectangle calculator?

A golden rectangle is a rectangle whose long side divided by its short side equals the golden ratio φ = (1 + √5)/2, about 1.6180. With the ratio fixed, a single measurement settles the whole shape, as it does for a square. φ has unusual arithmetic: φ² = φ + 1, about 2.618, and 1/φ = φ − 1, about 0.618, so the number, its square and its reciprocal share the same decimals. Ratios of consecutive Fibonacci numbers — 5/3, 8/5, 13/8, 21/13 — close in on φ from alternate sides, which is why 8 by 13 and 13 by 21 serve as whole-number stand-ins.

How to use this calculator.

1. 01Choose the measurement you already have: short side, long side, area, perimeter or diagonal.
2. 02Enter its value in any unit — millimetres, inches, pixels. Lengths come back in that unit and the area in that unit squared.
3. 03If the dimension you know should be the longer one, such as the width of a landscape banner, choose long side; if it should be the shorter, choose short side.
4. 04To test a rectangle you already have, enter its short side and compare the long side returned with the one you measure. A difference larger than your measuring error means it is not golden.
5. 05For the classic subdivision, mark a square on the short side; the leftover strip's short side is given, and you repeat on each new strip.

The formula.

With long side a, short side b and φ = (1 + √5)/2: a = bφ, area A = ab = b²φ, perimeter P = 2(a + b) = 2b(φ + 1), and diagonal d = √(a² + b²) = b√(φ² + 1), roughly 1.9021b. Each input is first turned into the short side: b = a/φ from the long side, b = √(A/φ) from the area, b = P/(2(φ + 1)) from the perimeter, b = d/√(φ² + 1) from the diagonal. Everything else is then rebuilt from b. Removing a b-by-b square leaves a rectangle measuring b by a − b, and a − b = b(φ − 1) = b/φ, because dividing φ² = φ + 1 through by φ gives φ − 1 = 1/φ. The leftover's sides are therefore in the ratio b to b/φ, which is φ again: the same shape at 1/φ of the size. Every length scales with b, so a 1% error in the measurement you enter gives 1% errors in the other lengths but about 2% in the area.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Golden Rectangle’ (live revision retrieved 2026-07-29). Equation (2) derives φ = (FC)/(CD) = (x(√5 + 1))/(2x) = ½(√5 + 1), and the entry states the defining self-similarity: partitioning the rectangle into a square and a new rectangle leaves a rectangle with sides in ratio 1:φ. Open access.
2. [2]Wolfram Research, MathWorld — ‘Golden Ratio’ (live revision retrieved 2026-07-29). Gives the defining quadratic φ² − φ − 1 = 0 and the decimal expansion 1.618033988749894848204586834365638117720… (cited there to OEIS A001622), against which this page's computed value was checked digit for digit. Open access.
3. [3]Euclid, Elements, Book VI, Definition 3 — ‘A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less’ — with the construction at Book VI, Proposition 30 (‘To cut a given finite straight line in extreme and mean ratio’). David E. Joyce web edition of the Heath translation, Clark University; both quotations confirmed on retrieval 2026-07-29. Used as the independent geometric derivation of φ.
4. [4]Markowsky, George. ‘Misconceptions about the Golden Ratio.’ The College Mathematics Journal 23, no. 1 (1992): 2–19, DOI 10.1080/07468342.1992.11973428. Peer-reviewed survey measuring the standard claimed occurrences of φ — the Parthenon, the Great Pyramid, the Mona Lisa, human body proportions — and showing that they do not hold up. Publisher-gated; abstract and bibliographic record open. Cited for this page's scope limit, not for any computed value.