Golden Rectangle Calculator
Calculate golden rectangle dimensions, area, and properties using the golden ratio φ ≈ 1.618
Calculate Golden Rectangle Properties
The shorter side of the golden rectangle
Golden Rectangle Properties
Golden Ratio Verification
Key Formulas
φ = (1 + √5) / 2 ≈ 1.618034
Length = Width × φ
Width = Length / φ
Segment b = Length - Width
Area = Width × Length
Golden Ratio Property
(a + b) / a = a / b = φ
Where a is width, b is segment, and φ is the golden ratio
This ratio creates the most aesthetically pleasing rectangle
Example Calculation
Golden Rectangle Art Frame
Problem: Design a golden rectangle picture frame
Given: Width (a) = 20 cm
Find: Length, segment b, and area
Solution Steps
1. Golden ratio: φ = (1 + √5) / 2 ≈ 1.618034
2. Length: a + b = a × φ = 20 × 1.618034 ≈ 32.36 cm
3. Segment b: b = (a + b) - a = 32.36 - 20 = 12.36 cm
4. Area: A = a × (a + b) = 20 × 32.36 = 647.2 cm²
Verification
Ratio check: (a + b) / a = 32.36 / 20 = 1.618 ✓
Ratio check: a / b = 20 / 12.36 = 1.618 ✓
Both ratios equal φ, confirming it's a true golden rectangle
Golden Rectangle Components
Width
The shorter side of the rectangle
Usually denoted as 'a'
Length
The longer side of the rectangle
Equal to width × φ
Segment b
Difference between length and width
b = (a + b) - a
Golden Ratio
φ = (1 + √5) / 2
≈ 1.618033988749
How to Draw a Golden Rectangle
Step 1
Construct a square (this becomes width 'a')
Step 2
Find the midpoint of one side
Step 3
Draw arc from midpoint to opposite vertex
Step 4
Extend square to arc intersection point
Fun Fact: The golden rectangle appears in nature, art, and architecture. Examples include the Parthenon, nautilus shells, and many classical paintings.
Understanding the Golden Rectangle
What is the Golden Rectangle?
A golden rectangle is a rectangle whose sides are in the golden ratio, approximately 1:1.618. This ratio is considered the most aesthetically pleasing proportion in visual design and appears frequently in nature, art, and architecture.
Mathematical Definition
- •Ratio of length to width equals the golden ratio φ
- •φ = (1 + √5) / 2 ≈ 1.618033988749
- •Satisfies the equation: (a + b) / a = a / b = φ
- •Self-similar: removing a square leaves another golden rectangle
Properties & Applications
Art & Design
Used in paintings, photography composition, and graphic design
Architecture
Found in the Parthenon, pyramids, and modern buildings
Nature
Appears in flower petals, nautilus shells, and spiral galaxies
Historical Note: The golden rectangle was known to ancient Greeks and Egyptians, who believed it represented divine proportion and perfect beauty.