How this calculator works
A golden rectangle is a rectangle whose sides divide in the golden ratio: the long side divided by the short side equals about 1.618. Removing a square from one end leaves a smaller rectangle with the same proportions, and the process can continue forever. Because all golden rectangles are similar, one measurement fixes every other one. The calculator takes the value you know, recovers the short side, and multiplies by the golden ratio to build the long side and the rest of the results.
Formula
Let φ = (1 + √5)/2 ≈ 1.6180339887. Then b = φa, a = b/φ = b(φ - 1), a = (a + b)/(1 + φ), a = diagonal/√(1 + φ²), and a = √(area/φ). Area = φa²; perimeter = 2φa; diagonal = a√(1 + φ²).
Worked example
A short side of 5 m gives a long side of 5φ ≈ 8.0902 m, total length ≈ 13.0902 m, area ≈ 40.4508 m², perimeter ≈ 26.1803 m and diagonal ≈ 9.5106 m.
Advertisement
Frequently asked questions
What makes a rectangle golden?
The ratio of its longer side to its shorter side must equal the golden ratio. Equivalently, the ratio of the total length to the longer side is the same number.
Is the golden rectangle the same as the golden spiral?
The golden spiral is built from a sequence of golden rectangles by adding quarter circles inside each square. The rectangle itself is the starting proportion; the spiral is a curve drawn through it.
Can I use the area to find the sides?
Yes. The area is φa², so the short side is the square root of the area divided by φ. The long side then follows by multiplying by φ.
Why does the diagonal use √(1 + φ²)?
The diagonal is the hypotenuse of a right triangle with legs a and φa. Factoring out a leaves √(1 + φ²), which is about 1.9021.