Golden Ratio Calculator

The golden ratio, denoted by the Greek letter phi (φ), is approximately 1.61803398875. It is defined mathematically as (1 + sqrt(5)) / 2. When a line is divided into two segments such that the ratio of the whole to the longer segment equals the ratio of the longer segment to the shorter segment, those segments are in the golden ratio.

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, ...) is formed by adding the previous two numbers. As you move further along the sequence, the ratio of consecutive Fibonacci numbers approaches the golden ratio. For example, 21/13 ≈ 1.615, 34/21 ≈ 1.619, 55/34 ≈ 1.618.

A golden rectangle is a rectangle whose sides are in the golden ratio. If the longer side is φ times the shorter side, the rectangle is golden. A unique property: if you remove a square from a golden rectangle, the remaining rectangle is also golden. This creates a spiral called the Fibonacci spiral.

The golden ratio appears in many natural structures: the spiral of a nautilus shell, the arrangement of leaves on plants, the proportions of the human face and body, spiral galaxies, and the branching of trees. It also appears frequently in art, architecture, and design because the human eye finds these proportions aesthetically pleasing.

Designers and architects use the golden ratio to create visually balanced compositions. It helps determine proportions of buildings, layouts of web pages, dimensions of photographs, and spacing in graphic design. The ratio naturally guides the human eye and creates a sense of harmony and balance.