Golden Ratio Calculator

The formula

φ = (1 + √5) / 2 ≈ 1.6180339887

The self-similar rectangle

Cut a square off a golden rectangle and what remains is another golden rectangle. That property defines φ: it is the number satisfying φ = 1 + 1/φ, or equivalently φ² = φ + 1. Solving gives (1 + √5)/2.

Repeating the cut forever traces the logarithmic spiral often drawn over nautilus shells. Claims that φ governs classical architecture and painting are mostly retrofitted — the mathematics is genuine, the art history much less so.

Area scales with the square

Double every length in a flat shape and the area quadruples; triple them and it grows ninefold. Perimeter, being a length, scales in step with the sides. That mismatch is why bigger shapes enclose proportionally more area per unit of edge — and why a circle, having no corners at all, encloses the most area for a given perimeter of any shape.

Units

Whatever unit you enter, the area comes out in that unit squared and the perimeter in the unit itself. The calculator is unit-agnostic — enter centimetres and read cm and cm², or feet and read ft and ft². Just do not mix units between fields.