Golden Ratio Calculator
Calculate the lengths of segments that form the golden ratio (approximately 1.618033988749895). Input the total length of a segment to find the lengths of the two parts that satisfy the golden ratio, with step-by-step explanations and visual representations.
Result
Detailed Steps
Visual Representation
Formulas
Golden Ratio Definition
The golden ratio (φ) is defined as:
Segment Division
For a segment of length L, the golden ratio divides it into two parts, a (longer) and b (shorter), such that:
Thus:
How to Use the Golden Ratio Calculator
- Enter the total length of the segment in the input field (e.g., 100).
- Optionally, adjust the precision level for decimal calculations (default is 5).
- Choose your preferred display mode (Standard, Step by Step, or Chart).
- Click the “Calculate” button.
- View your results in your chosen display format, showing the lengths of the two segments.
Frequently Asked Questions (FAQs)
What is the golden ratio?
The golden ratio (φ) is an irrational number approximately equal to 1.618033988749895. It is defined as the ratio where the larger part divided by the smaller part equals the whole divided by the larger part, i.e., (a + b)/a = a/b = φ.
How do you calculate the golden ratio segments?
For a segment of length L, the longer part (a) is calculated as a = L / (1 + 1/φ), and the shorter part (b) is b = L – a. This ensures a/b = φ.
Where is the golden ratio used?
The golden ratio appears in mathematics, art, architecture, and nature. It is used in design to create aesthetically pleasing proportions, in geometry for constructions like the golden rectangle, and is observed in patterns like the Fibonacci sequence and certain natural spirals.
Is the golden ratio exact?
No, the golden ratio is an irrational number, so its decimal representation is infinite and non-repeating. This calculator uses a high-precision approximation (φ ≈ 1.618033988749895) and allows you to set the desired decimal precision for results.