Solve every dimension of a golden rectangle — short side, long side, area,
perimeter and diagonal — from any single known value, with the exact ratio
b/a = φ = (1+√5)/2 ≈ 1.6180339887.
Includes a live nested-squares + golden-spiral diagram. Everything runs in your browser; nothing is uploaded.
Dimension solver
Pick what you know, enter the value, get the other four dimensions.
Short side a
0units
Long side b
0units
Area A
0units²
Perimeter P
0units
Diagonal d
0units
Check b/a
0≈ φ
Nested squares & golden spiral
Whirling-squares subdivision of your rectangle, with the quarter-circle golden spiral. Redraws live.
Worked examples
Computed from the frozen oracle vectors (10 decimals) — never hand-typed.
1 — Short side a = 10
a = 10.0000000000; b = 16.1803398875; A = 161.8033988750; P = 52.3606797750; d = 19.0211303259; check b/a = 1.6180339887.
2 — Long side b = 100
a = 61.8033988750; b = 100.0000000000; A = 6180.3398874989; P = 323.6067977500; d = 117.5570504585; check b/a = 1.6180339887.
3 — Diagonal d = 50
a = 26.2865556060; b = 42.5325404176; A = 1118.0339887499; P = 137.6381920471; d = 50.0000000000; check b/a = 1.6180339887.
4 — Area A = 1000
a = 24.8602893939; b = 40.2247932095; A = 1000.0000000000; P = 130.1701652069; d = 47.2870804502; check b/a = 1.6180339887.
Methodology
b = a·φ A = a·b P = 2(a+b) d = a·√(1+φ²)
Constants: φ = (1+√5)/2 = 1.618033988749895… (full double precision, not 1.618);
√(1+φ²) = 1.902113032590307…. Inverse modes: a = b/φ, a = d/√(1+φ²),
a = √(A/φ). Honest note: the diagonal factor is √(1+φ²), not √(φ³)
— 1+φ² ≈ 3.6180340 while φ³ ≈ 4.2360680, a mistake found in some sources
(and in our own first draft, caught by the oracle).
Scope boundary: what this tool is (and isn't)
This tool = golden RECTANGLE dimensions. It answers “given one of a/b/A/P/d, what are the
rest?” for the rectangle with b/a = φ, and draws its nested squares + spiral.
NOT the number φ itself. For the golden ratio's value, formulas and properties, use
golden-ratio-calc (legacy).
A golden rectangle is a rectangle whose long side divided by its short side equals the golden ratio φ = (1+√5)/2 ≈ 1.6180339887. Its defining property: cut off the largest possible square and the remaining rectangle is itself a golden rectangle, so the subdivision repeats forever. This page computes every dimension of such a rectangle — sides, area, perimeter and diagonal — from any single known value.
What is the exact value of the golden ratio φ?
φ = (1+√5)/2 = 1.618033988749895…. It is irrational: the digits never repeat. This calculator uses full double-precision φ (not 1.618), so b = a·φ is exact to about 15 significant digits — for example a = 10 gives b = 16.1803398875, not 16.18.
How do I find the long side from the short side?
Multiply by φ: b = a·φ. With a = 10 units the long side is b = 16.1803398875, the area is A = 161.8033988750, the perimeter is P = 52.3606797750 and the diagonal is d = 19.0211303259. Enter the short side in the short-side mode above to get all four at once.
How do I find the short side from the long side?
Divide by φ: a = b/φ. A golden rectangle with long side 100 has short side a = 61.8033988750, area 6180.3398874989 and diagonal 117.5570504585. The ratio b/a always returns exactly φ, which the results panel verifies for every calculation.
How do I get the sides from the diagonal?
Since d = √(a²+b²) = a·√(1+φ²), the short side is a = d/√(1+φ²) with √(1+φ²) ≈ 1.9021130326. For d = 50: a = 26.2865556060 and b = 42.5325404176 (check: √(26.2865556060² + 42.5325404176²) = 50.0000000000 to ten decimals). Warning: some sources write d = a·√(φ³) — that is wrong: √(φ³) ≈ 2.0581710273 while √(1+φ²) ≈ 1.9021130326.
How do I get the sides from the area?
From A = a·b = a²·φ, the short side is a = √(A/φ) and the long side b = √(A·φ). For A = 1000: a = 24.8602893939, b = 40.2247932095, perimeter 130.1701652069 and diagonal 47.2870804502. Every value on this page is recomputed from the frozen oracle vectors, never hand-typed.
What is the diagonal of a golden rectangle?
d = a·√(1+φ²) ≈ 1.9021130326·a — about 1.9021 times the short side. For a = 10 the diagonal is 19.0211303259. Because φ² = φ + 1, you can also write √(1+φ²) = √(φ+2), but not √(φ³): 1+φ² ≈ 3.6180340 while φ³ ≈ 4.2360680.
What are the nested squares in the diagram?
They visualise the defining property: each golden rectangle splits into a square plus a smaller golden rectangle. Square sides shrink by exactly 1/φ each step (10, 6.1803398875, 3.8196601125, …). The diagram above draws the first seven squares for your rectangle, recomputed live from your input.
What is the spiral in the diagram?
Each square carries a quarter-circle arc; chained together with matching tangents they approximate the golden (logarithmic) spiral r = c·φ^(2θ/π). Toggle the spiral and the squares independently above the diagram.
How is this different from the golden ratio and golden spiral tools?
Scope boundary: golden-ratio-calc covers the number φ itself (value, formulas, properties). golden-spiral-dimension-calculator covers spiral geometry. This page is the rectangle dimension solver: sides, area, perimeter and diagonal from any one known value, plus the nested-squares diagram. The three pages link to each other and do not duplicate functionality.
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