Golden Ratio
Ratio φ where the whole is to the larger part as the larger is to the smaller.
Key idea
Ratio φ where the whole is to the larger part as the larger is to the smaller.
In depth
φ = (1 + √5)/2 ≈ 1.618 satisfies φ² = φ + 1. A line divided in golden ratio has whole : larger = larger : smaller. It appears in the regular pentagon, Fibonacci ratios and spirals.
Worked example
Fibonacci ratio 55/34 ≈ 1.6176, close to φ.
Formula
Real-world uses
- Proportions in art, architecture and logo design
- The spiral pattern in shells and flowers
Common mistake
Believing it is the key to beauty. Many claims about it are exaggerated.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Golden Ratio?
Ratio φ where the whole is to the larger part as the larger is to the smaller. It is a topic at the advanced level in Geometry.
What is the formula for Golden Ratio?
The key formula is φ = (1+√5)/2 ≈ 1.618.
Where is Golden Ratio used in real life?
Real-world examples: Proportions in art, architecture and logo design; The spiral pattern in shells and flowers.
What should I learn before Golden Ratio?
Make sure you are comfortable with Geometric Probability, the previous topic in Geometry.
What is a common mistake with Golden Ratio?
Believing it is the key to beauty. Many claims about it are exaggerated.
How can I practise Golden Ratio?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Golden Ratio?
Move on to Fractals next.
Related advanced topics in Geometry
- Heron’s Formula – Triangle area from three sides.
- Regular Polygon Area – Use apothem and perimeter.
- Arc Length of a Circle – Part of the circumference.
- Sector Area – Part of the circle’s area.
- Central and Inscribed Angles – An inscribed angle is half the central angle on the same arc.
- Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.