Golden Ratio Calculator
Input one dimension to calculate other dimensions according to the golden ratio.
Golden Ratio (φ)
1.618034
A:B Ratio
1.618:1
(A+B):A Ratio
1.618:1
Fibonacci Sequence
1
1
2
3
5
8
13
21
34
55
89
144
233
The ratio of consecutive Fibonacci numbers converges to the golden ratio. (e.g., 144/89 ≈ 1.618)
Complete Golden Ratio Guide
✨ What is the Golden Ratio?
The Golden Ratio (φ) is a special mathematical ratio approximately 1.618033988... When two quantities A and B satisfy (A+B):A = A:B, they are in golden ratio. Considered the most aesthetically pleasing proportion since ancient Greece.
🔢 Mathematical Definition
• φ = (1 + √5) / 2 ≈ 1.618033988749895
• φ² = φ + 1
• 1/φ = φ - 1 ≈ 0.618
• Golden ratio is an irrational, non-repeating infinite decimal
🌀 Relationship with Fibonacci Sequence
In the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...), the ratio of consecutive numbers converges to the golden ratio:
• 5/3 = 1.666...
• 8/5 = 1.6
• 13/8 = 1.625
• 144/89 = 1.61798...
• 233/144 = 1.618055... (almost exactly φ)
As the sequence progresses, the ratio approaches exactly φ = 1.618033...
🎨 Applications in Design and Art
• Famous Paintings: Leonardo da Vinci's Mona Lisa, The Last Supper
• Logo Design: Apple, Twitter, Pepsi logos
• Web Design: Content to sidebar ratio (960px : 593px)
• Typography: Heading to body font size ratio (26pt : 16pt)
• Layout: Credit card ratio (86mm × 54mm ≈ 1.59)
🏛️ Golden Ratio in Architecture
• Parthenon: Front width to height ratio
• Pyramids: Base to height relationship
• Modern Architecture: UN Headquarters, Toronto CN Tower
• Interior Design: Room dimensions, furniture placement, frame proportions
🌿 Golden Ratio in Nature
• Plants: Sunflower seed spirals, pinecone scales, pineapple skin
• Animals: Nautilus shell spirals, butterfly wings
• Human Body: Upper to lower body ratio, finger joint lengths
• DNA: Width and rotation of double helix
💡 Practical Tips
• Poster Design: Height 1000mm → Width 618mm
• Business Cards: 90mm × 55.6mm
• Photo Composition: Place subject at golden division points
• PPT Layout: Title to content area ratio
• Book Pages: Margin to text area ratio
The Golden Ratio (φ) is a special mathematical ratio approximately 1.618033988... When two quantities A and B satisfy (A+B):A = A:B, they are in golden ratio. Considered the most aesthetically pleasing proportion since ancient Greece.
🔢 Mathematical Definition
• φ = (1 + √5) / 2 ≈ 1.618033988749895
• φ² = φ + 1
• 1/φ = φ - 1 ≈ 0.618
• Golden ratio is an irrational, non-repeating infinite decimal
🌀 Relationship with Fibonacci Sequence
In the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...), the ratio of consecutive numbers converges to the golden ratio:
• 5/3 = 1.666...
• 8/5 = 1.6
• 13/8 = 1.625
• 144/89 = 1.61798...
• 233/144 = 1.618055... (almost exactly φ)
As the sequence progresses, the ratio approaches exactly φ = 1.618033...
🎨 Applications in Design and Art
• Famous Paintings: Leonardo da Vinci's Mona Lisa, The Last Supper
• Logo Design: Apple, Twitter, Pepsi logos
• Web Design: Content to sidebar ratio (960px : 593px)
• Typography: Heading to body font size ratio (26pt : 16pt)
• Layout: Credit card ratio (86mm × 54mm ≈ 1.59)
🏛️ Golden Ratio in Architecture
• Parthenon: Front width to height ratio
• Pyramids: Base to height relationship
• Modern Architecture: UN Headquarters, Toronto CN Tower
• Interior Design: Room dimensions, furniture placement, frame proportions
🌿 Golden Ratio in Nature
• Plants: Sunflower seed spirals, pinecone scales, pineapple skin
• Animals: Nautilus shell spirals, butterfly wings
• Human Body: Upper to lower body ratio, finger joint lengths
• DNA: Width and rotation of double helix
💡 Practical Tips
• Poster Design: Height 1000mm → Width 618mm
• Business Cards: 90mm × 55.6mm
• Photo Composition: Place subject at golden division points
• PPT Layout: Title to content area ratio
• Book Pages: Margin to text area ratio
Complete Golden Ratio Guide (2025)
01
Mathematical Definition and φ (Phi) Value
The Golden Ratio (φ) is known as the most beautiful ratio in mathematics. Its precise value is φ = (1 + √5) / 2 = 1.618033988749895..., an irrational number with infinite non-repeating decimals. When two quantities a and b satisfy (a+b):a = a:b, they are in golden ratio. Solving this equation gives x² = x + 1, with φ as the positive solution. The golden ratio has remarkable mathematical properties: φ² = φ + 1 = 2.618..., 1/φ = φ - 1 = 0.618.... Adding 1 to φ gives its square, and its reciprocal equals φ minus 1. Ancient Greek mathematician Euclid called this the "extreme and mean ratio" in Elements. The golden ratio naturally appears in regular pentagons and pentagrams - the ratio of diagonal to side equals φ. Renaissance mathematician Luca Pacioli considered it divine providence in his work "De Divina Proportione".
02
Fibonacci Sequence and Golden Ratio Relationship
The Fibonacci Sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...) creates each number by adding the previous two. Remarkably, the ratio of consecutive numbers converges to φ. For example: 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615..., 34/21 = 1.619..., 89/55 = 1.618181..., 144/89 = 1.617977..., 233/144 = 1.618055... approaching φ = 1.618033... This is mathematically proven by the general term formula F(n) = (φⁿ - (-φ)⁻ⁿ) / √5. Nature demonstrates this relationship: sunflower seeds arrange in 34 clockwise and 55 counterclockwise spirals (consecutive Fibonacci numbers). Pinecones show 8 and 13 spiral patterns. This occurs because plants use the golden angle (137.5°) for optimal seed packing.
03
Golden Ratio in Design and Layout
Graphic design uses the golden ratio as a key tool for visual harmony. Web design: A 960px wide site can have content area 593px (960÷1.618) and sidebar 367px (593÷1.618) for natural layout. Twitter, Apple, and Pepsi logos are all based on golden ratio circles. Typography: With 16pt body text, use 26pt (16×1.618) for headings and 10pt (16÷1.618) for captions to create balanced hierarchy. Business cards: Standard cards (90mm × 54mm) approximate the golden ratio (90÷54=1.67). Posters: A 1000mm tall poster should be 618mm wide. Photo composition: Placing subjects at golden division points (more precise than rule of thirds) naturally draws the eye. Margins: Dividing book pages with φ ratio between margins and text improves readability. Adobe Photoshop and Illustrator include built-in golden ratio guide templates.
04
Golden Ratio in Architecture and Art
Ancient architecture: The Parthenon in Athens (447-432 BC) has front width to height ratios approximating φ. Egypt's Great Pyramid of Giza (2580-2560 BC) shows φ in the ratio of half base to height. Modern architecture: Le Corbusier developed the "Modulor" system combining human proportions (183cm) with golden ratios. UN Headquarters (New York), Toronto CN Tower, and Seoul 63 Building all feature φ proportions. Paintings: Leonardo da Vinci's Mona Lisa (1503-1519) follows golden ratios in facial contours and feature placement. The Last Supper (1495-1498) uses golden rectangle structure for table and figure arrangement. Michelangelo's Creation of Adam (1508-1512) positions key composition lines at golden division points. Modern art: Piet Mondrian's Composition with Red, Yellow and Blue (1930) uses golden ratio grids. Salvador Dali intentionally used golden ratio canvases for "The Sacrament of the Last Supper" (1955).
05
Golden Ratio in Nature (Spirals, Petals)
Plant spirals: Sunflower seeds arrange from center outward rotating by golden angle 137.5° (360° ÷ φ²), maximizing space usage. Pinecones, pineapples, and cactus spines show 8/13 or 13/21 spiral patterns (Fibonacci numbers). Petal counts: Lilies 3, buttercups 5, cosmos 8, marigolds 13, asters 21, daisies 34, 55, 89 - all Fibonacci numbers. This occurs because growth points generate petals at golden angles. Tree branches: Many trees show φ ratios between trunk and branch lengths and node spacing. Animals: Nautilus shells form golden spirals, growing by φ every 90° rotation. Similar patterns appear in butterfly wings and snail shells. Human body: Total height to navel height averages 1.618. Finger joint lengths (fingertip-second joint-palm base) show φ ratios. DNA double helix width (34Å) to length per turn (21Å) gives 34/21 = 1.619.
06
Practical Web/UI Design with Golden Ratio
Layout grids: Divide screen width by φ for main and secondary areas. Example: 1920px screen → main 1187px (1920÷1.618), sidebar 733px. Mobile (375px): main 232px, margin 143px. Button sizing: A 44px tall button with 71px (44×1.618) width creates balanced CTAs. Card design: For 300px card height, use 185px (300÷1.618) image area and 115px (185÷1.618) text area. Font scale: From 16px base → 12px (÷1.33), 16px, 21px (×1.33), 28px, 37px, 49px, 65px system (1.33 = √φ). Spacing system: 8px, 13px, 21px, 34px, 55px, 89px (Fibonacci). Color ratios: Primary 60%, secondary 30%, accent 10% (60:30 = 2:1, 30:10 = 3:1, Fibonacci ratios). Responsive breakpoints: 320px, 518px (320×φ), 838px (518×φ), 1356px. Animation timing: 0.1s, 0.162s (0.1×φ), 0.262s - natural motion. CSS aspect-ratio: 1.618 / 1 creates golden ratio containers easily. Figma and Sketch have Golden Ratio Calculator plugins for real-time application.