Golden Rectangle Calculator

Background.

The Quanta golden rectangle calculator takes any single measurement of a rectangle whose sides stand in the golden ratio and returns every other measurement it has. Enter the short side, the long side, the area, the perimeter or the diagonal, and the calculator returns both sides, the area, the perimeter, the diagonal, the golden ratio itself, and the short side of the smaller golden rectangle that remains once you cut a square off one end. With the default short side of 10, the long side comes back as 16.1803398875, the area as 161.803398875, the perimeter as 52.360679775 and the diagonal as 19.0211303259.

A golden rectangle has exactly one degree of freedom. Ordinary rectangles need two numbers to pin them down, because the length and the width are independent; here the ratio between them is fixed at φ = (1 + √5) ÷ 2 = 1.6180339887, so a single measurement determines the shape completely. That is why this page asks for one number rather than two, and it is the difference between this calculator and a general area-and-perimeter tool: those need both sides and cannot supply a missing one, because they have no ratio to work from.

The definition goes back to Euclid, who did not call it golden. Book VI, Definition 3 of the Elements describes a line "cut in extreme and mean ratio" — divided so that the whole is to the greater part as the greater part is to the lesser — and Book VI, Proposition 30 gives the straightedge-and-compass construction. Written algebraically that condition is (a + b) ÷ a = a ÷ b, which rearranges to φ² − φ − 1 = 0 and has the single positive root (1 + √5) ÷ 2. Everything else on this page follows from that one quadratic.

The property that makes the shape genuinely interesting rather than merely proportioned is self-similarity. Remove a square whose side equals the short side, and the rectangle left behind is again golden, at a scale reduced by exactly one factor of φ. Repeat the cut on the leftover and you get another, and another, without end. Drawing a quarter-circle inside each square in turn produces the familiar spiral. This calculator reports the short side of that first leftover rectangle so you can start the sequence, and the algebra behind it is a restatement of the same quadratic: a − b = b ÷ φ, because φ − 1 = 1 ÷ φ.

One caveat belongs beside the answer rather than three screens below it. The mathematics of the golden ratio is exact and uncontroversial; the claims routinely made about where it appears are not. George Markowsky's 1992 survey in the College Mathematics Journal measured the usual examples — the Parthenon, the Great Pyramid, the Mona Lisa, the proportions of the human body, the dimensions of a credit card — and found that none of them match φ under any consistent choice of what to measure from and to, and that several of the standard citations trace back to sources that never made the claim at all. Use this page to build a golden rectangle, to check the geometry of one, or to lay out a design you have decided you want in that proportion. Do not use it as evidence that a rectangle in the wild is golden, because the odds are that it is not, and the honest reason to choose the proportion is that you like 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 ≈ 1.6180339887. Wolfram MathWorld's Golden Rectangle entry derives that value in its Equation (2) as φ = (x(√5 + 1)) ÷ (2x) = ½(√5 + 1), and characterises the shape by its defining behaviour: partitioning the rectangle into a square and a new rectangle leaves a new rectangle whose sides are again in the ratio 1 : φ. The underlying proportion is Euclid's, from Elements Book VI Definition 3 — a line is cut in extreme and mean ratio when the whole is to the greater segment as the greater is to the lesser — with the construction given in Book VI Proposition 30. Setting φ = a ÷ b in that definition gives (a + b) ÷ a = a ÷ b, hence φ² = φ + 1, whose positive root is (1 + √5) ÷ 2. Two consequences of that quadratic do most of the work on this page. First, φ − 1 = 1 ÷ φ, so the leftover rectangle after removing the square is scaled down by exactly one factor of φ and is therefore golden itself, infinitely often. Second, φ is irrational, which means no golden rectangle has both sides rational: pick a whole number for one side and the other is an unending decimal. The successive ratios of Fibonacci numbers — 3/2, 5/3, 8/5, 13/8, 21/13 — converge to φ, which is why 8 by 13 and 13 by 21 turn up as practical approximations in design work and in graph-paper constructions.

How to use this calculator.

1. 01Choose which measurement you already have from the dropdown: short side, long side, area, perimeter or diagonal. A golden rectangle has one degree of freedom, so any one of these fixes the whole shape.
2. 02Enter that value in any unit you like. Millimetres, inches, pixels and metres all work — lengths come back in the same unit and the area comes back in that unit squared.
3. 03Read the long side at the top. In short-side mode it is the number you came for: the partner side that makes the rectangle golden.
4. 04Read the area, perimeter and diagonal underneath if you need them for a layout, a material estimate or a frame.
5. 05Read the leftover rectangle's short side to start the classic subdivision. Cut a square off the golden rectangle and the piece that remains is golden too, with that short side.
6. 06To check whether a rectangle you already have is golden, type its short side and compare the long side the calculator returns with the one you measured. If they differ by more than your measuring error, the rectangle is not golden — and given how many objects are wrongly claimed to be, that is the expected outcome.

The formula.

Every mode on this page solves for the short side b first, then rebuilds the rest of the rectangle from it. Short-side mode is trivial. Long-side mode divides by φ, since a = bφ. Area mode uses A = ab = b²φ, so b = √(A ÷ φ). Perimeter mode uses P = 2(a + b) = 2b(φ + 1), so b = P ÷ (2(φ + 1)). Diagonal mode uses d = √(a² + b²) = b√(φ² + 1), so b = d ÷ √(φ² + 1). Once b is known, a = bφ, A = ab, P = 2(a + b) and d = √(a² + b²).

Working the default all the way through: b = 10, so a = 10 × 1.618033988749894848 = 16.180339887498948, displayed as 16.1803398875. The area is 16.180339887498948 × 10 = 161.80339887498948, displayed as 161.803398875. The perimeter is 2 × (16.180339887498948 + 10) = 52.360679774997896, displayed as 52.360679775. The diagonal is √(16.180339887498948² + 10²) = √(261.803398874989484 + 100) = √361.803398874989484 = 19.021130325903071, displayed as 19.0211303259. The leftover rectangle's short side is 16.180339887498948 − 10 = 6.180339887498948, displayed as 6.1803398875 — and notice that it equals 10 ÷ 1.618033988749894848 to every digit shown, which is the identity φ − 1 = 1 ÷ φ in action.

Why that identity holds: the defining condition φ² = φ + 1 can be divided through by φ to give φ = 1 + (1 ÷ φ), so φ − 1 = 1 ÷ φ. Geometrically it says the square you removed shrank the rectangle by exactly one factor of φ, which is the reason the subdivision never terminates and never changes shape.

Rounding stage: every intermediate is carried at 40 significant digits in Decimal arithmetic and rounded once, at the return boundary, to 10 decimal places. φ itself is computed as (1 + √5) ÷ 2 at that precision rather than entered as the literal 1.618033989 — a literal would inject error into the ninth decimal place of the area, which is quadratic in the side length. The short side is never rounded before the long side, area, perimeter and diagonal are derived from it. Nothing on this page is sorted into bands or graded, so no threshold depends on display rounding.

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.