Golden Rectangle Calculator

Calculate the dimensions of a golden rectangle from one side using the golden ratio phi. Explore area, perimeter, diagonal, and spiral construction.

Enter a Side Length

Result

Golden Rectangle Dimensions
16.18 x 10
Long Side (a) 16.18033989
Short Side (b) 10
Ratio (a/b) 1.61803399
Area 161.803399
Perimeter 52.360680
Diagonal 19.021130
Inner Square Side 10
Remaining Rectangle 6.18034 x 10

Step-by-Step Solution

a = b x phi = 10 x 1.618034 = 16.18034

What Is a Golden Rectangle?

A golden rectangle is a rectangle whose side lengths are in the golden ratio, approximately 1:1.618. When you remove a square from a golden rectangle (with the square's side equal to the shorter side of the rectangle), the remaining rectangle is also a golden rectangle. This self-similar property continues infinitely and gives rise to the famous golden spiral.

Properties of the Golden Rectangle

Dimensions

The long side equals the short side multiplied by phi.

a = b x phi = b x 1.618...

Self-Similarity

Removing a square leaves another golden rectangle.

Remaining: (a - b) x b is golden

Area

Area equals short side squared times phi.

Area = b^2 x phi

Diagonal

Use the Pythagorean theorem with golden sides.

d = b x sqrt(1 + phi^2)

The Golden Spiral

By repeatedly subdividing a golden rectangle into squares and smaller golden rectangles, then drawing a quarter-circle arc in each square, you create the golden spiral. This logarithmic spiral approximation appears throughout nature, from nautilus shells to hurricane formations to spiral galaxies.

How to Construct a Golden Spiral

  1. Start with a golden rectangle with sides a and b (where a = b x phi).
  2. Draw a square with side length b inside the rectangle on the left or right.
  3. The remaining area is a new, smaller golden rectangle rotated 90 degrees.
  4. Repeat the process with the smaller rectangle.
  5. Draw quarter-circle arcs in each successive square to form the spiral.

Applications of the Golden Rectangle

  • Architecture: The Parthenon's facade, Le Corbusier's Modulor system, and many buildings use golden rectangles.
  • Art: Composition grids based on golden rectangles help artists create harmonious layouts.
  • Design: Business cards, book covers, and screen aspect ratios often approximate the golden ratio.
  • Photography: The "phi grid" (similar to the rule of thirds) uses golden ratio divisions.
  • Typography: Line width to font size ratios often follow golden proportions for readability.

Interesting Facts

A standard credit card measures approximately 85.6mm x 53.98mm, giving a ratio of about 1.586, which is very close to the golden ratio. Many common paper sizes and screen aspect ratios also approximate golden proportions, contributing to their perceived visual appeal.