Golden Ratio Calculator
Find values satisfying the golden ratio (A+B):A = A:B. Enter any term to calculate missing phi (φ) values with visual verification.
Calculate golden ratio proportions where (A+B):A = A:B = φ. Enter any segment to find the complete golden proportion set.
How to Use
Understanding Golden Ratio Calculations
This calculator finds values that satisfy the golden ratio relationship where (A+B):A = A:B. Enter any one value—the longer segment A, shorter segment B, or the whole length (A+B)—and the calculator determines the other two values that complete the golden proportion.
Three Calculation Modes:
- Given A (longer): Calculates B = A/φ and whole = A + B
- Given B (shorter): Calculates A = B × φ and whole = A + B
- Given whole (A+B): Calculates A = whole/φ and B = whole - A
Precision Options:
Results are displayed with up to 6 decimal places for precision. You can round values A and B to whole numbers for practical applications like design measurements, or keep the full decimal precision for mathematical accuracy.
What is Golden Ratio Calculator?
The golden ratio, also known as the golden mean or divine proportion, is represented by the Greek letter phi (φ). It occurs when a line is divided into two parts such that the ratio of the whole length to the longer segment equals the ratio of the longer segment to the shorter segment: (A+B)/A = A/B = φ.
The exact value of phi is (1 + √5) / 2, which equals approximately 1.6180339887499. This irrational number has fascinated mathematicians, artists, and scientists for millennia due to its unique mathematical properties and frequent appearance in nature.
The golden ratio has a special relationship with the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...). As you progress through the sequence, the ratio of consecutive Fibonacci numbers approaches phi. For example: 13/8 = 1.625, 21/13 ≈ 1.615, 34/21 ≈ 1.619—each getting closer to 1.618...
Formula
Golden Ratio Definition:
φ = (1 + √5) / 2 ≈ 1.6180339887499
The exact irrational value of phi
Golden Ratio Property:
(A + B) / A = A / B = φ
Both ratios equal phi when A and B are in golden proportion
Calculating Golden Ratio Values:
Given A: B = A / φ
Given B: A = B × φ
Given (A+B): A = (A+B) / φ
Unique Property of Phi:
φ² = φ + 1 ≈ 2.618
1/φ = φ - 1 ≈ 0.618
Phi is the only number where squaring adds 1 and reciprocal subtracts 1
Examples
Example 1: Given Longer Segment A = 10
B = A / φ = 10 / 1.618 = 6.180
Whole = A + B = 10 + 6.180 = 16.180
Golden ratio set: A = 10, B = 6.18, Whole = 16.18
Example 2: Given Shorter Segment B = 5
A = B × φ = 5 × 1.618 = 8.090
Whole = A + B = 8.090 + 5 = 13.090
Golden ratio set: A = 8.09, B = 5, Whole = 13.09
Example 3: Given Whole Length = 100
A = Whole / φ = 100 / 1.618 = 61.803
B = Whole - A = 100 - 61.803 = 38.197
Golden ratio set: A = 61.80, B = 38.20, Whole = 100
Example 4: Verification
For A = 10, B = 6.18:
(A + B) / A = 16.18 / 10 = 1.618 ✓
A / B = 10 / 6.18 = 1.618 ✓
Both ratios equal φ, confirming golden proportion