What Is the Golden Rectangle?
A golden rectangle is a rectangle whose side lengths are in the golden ratio φ (phi). Specifically, if the shorter side is a and the longer side is b, then:
b / a = φ = (1 + √5) / 2 ≈ 1.6180339887…
The golden rectangle is considered one of the most visually pleasing shapes in nature and art. You can find it in the Parthenon, Leonardo da Vinci's paintings, and even in the spiral of a nautilus shell.
How to Use This Golden Rectangle Calculator
- Choose what you know: width (shorter side a), length (longer side b), or the area.
- Enter the value.
- Select your unit — metric (mm, cm, m, km) or US imperial (in, ft, yd, mi).
- Click Calculate.
The calculator will instantly find the missing side and the area of your golden rectangle.
Golden Rectangle Formulas
- Golden ratio: φ = (1 + √5) / 2 ≈ 1.6180339887
- Given width (a): length b = a × φ
- Given length (b): width a = b / φ
- Area: A = a × b = a² × φ
How to Draw a Golden Rectangle
- Draw a square with side length a.
- Find the midpoint M of one side of the square.
- Draw an arc centered at M with radius equal to the distance from M to an opposite corner of the square.
- The point where this arc meets the extension of the base of the square gives you the corner of the golden rectangle.
- Complete the rectangle — its length is a × φ.
Interesting Properties
- If you cut a square from a golden rectangle, the remaining piece is another golden rectangle.
- Placing squares inside a golden rectangle in this recursive way traces out a golden spiral.
- The golden rectangle ratio appears in the proportions of the human face, credit cards, and many famous artworks.
Frequently Asked Questions
How do I find the width of a golden rectangle given the length?
Divide the length by the golden ratio: width = length / φ ≈ length / 1.618.
How do I find the length given the width?
Multiply the width by the golden ratio: length = width × φ ≈ width × 1.618.
What is the area of a golden rectangle?
Area = width × length = a × (a × φ) = a² × φ. Alternatively, if you know the length b: Area = (b / φ) × b = b² / φ.
Does the golden rectangle have any real-world applications?
Yes — architects, designers, and artists use the golden rectangle to create naturally pleasing proportions. It appears in screen aspect ratios, photography composition, logo design, and architecture.