Golden Rectangle Calculator — φ Dimensions & Area
Enter either the short or the long side of a rectangle and the calculator finds the other side using the golden ratio φ = (1+√5)/2 ≈ 1.6180, then gives the area, perimeter and diagonal.
Known side
Ratio of long side to short side — should equal (1+√5)/2 ≈ 1.618034
- 1
Golden ratio φ
(1 + √5) ÷ 2 = 1.618034 - 2
Long side b = a × φ
10 × 1.618034 = 16.18034 - 3
Ratio b ÷ a
16.18034 ÷ 10 = 1.618034
How does this calculator work?
A golden rectangle has sides in ratio φ = (1+√5)/2 ≈ 1.61803. Given the short side a, the long side is a × φ and the area is a² × φ. Enter either side and get the full set of dimensions, area, perimeter and diagonal — used in architecture, design and nature since antiquity.
Formula
How this is calculated
A golden rectangle is one where the ratio of the long side to the short side equals the golden ratio φ = (1+√5)/2 ≈ 1.61803. This number appears throughout geometry, art, and nature — the Parthenon facade, many Renaissance paintings, and the nautilus shell are all often cited (though some of these associations are approximate).
If you know the short side a, the long side is b = a × φ. If you know the long side b, the short side is a = b / φ. Once both sides are known, area = a × b, perimeter = 2(a + b), and diagonal = √(a² + b²) follow from standard rectangle formulas.
One elegant property: if you remove a square from a golden rectangle, the remaining piece is itself a golden rectangle, and you can repeat this process indefinitely. Drawing a quarter-circle in each successive square traces the Fibonacci spiral (or golden spiral), a logarithmic spiral that appears widely in plant phyllotaxis and other natural growth patterns.
Frequently asked questions
The golden ratio φ = (1+√5)/2 ≈ 1.61803 is an irrational number that arises when a line segment is divided so that the whole is to the larger part as the larger part is to the smaller. It appears in Fibonacci sequences, regular pentagons, and many natural growth patterns.
Deliberately in some cases (Le Corbusier explicitly used it in his Modulor system), approximately in others (the Parthenon proportions are close but not exact), and coincidentally in many more — with 1.618 being so close to common aesthetic preferences for rectangles that many correlations are confirmation bias.
The ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, …) converges to φ. So a rectangle whose sides are consecutive Fibonacci numbers (e.g. 8×13) is an excellent approximation of a golden rectangle, and successive square removals trace a near-perfect Fibonacci spiral.
Also known as
TG we-Calculate Editorial Team. (2026). Golden Rectangle Calculator — φ Dimensions & Area [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/golden-rectangle-calculator
TG we-Calculate Editorial Team. "Golden Rectangle Calculator — φ Dimensions & Area." TG we-Calculate. 2026. https://we-calculate.com/calculator/golden-rectangle-calculator.
TG we-Calculate Editorial Team, "Golden Rectangle Calculator — φ Dimensions & Area," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/golden-rectangle-calculator
@misc{wecalculate_golden_rectangle_calculator, title = {Golden Rectangle Calculator — φ Dimensions & Area}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/golden-rectangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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