QuickCalculators finds missing golden-ratio terms from any one known length among A, B, and A + B, using phi ≈ 1.6180339887 as the defining constant. Enter a single known value and read the other two terms with the exact phi-based formulas shown.
Find the missing golden ratio terms from any one value
Two positive lengths A and B form a golden ratio when (A + B) / A = A / B = phi. Golden Ratio Calculator solves for the unknown lengths once any one of A, B, or A + B is supplied. The larger part relates to the smaller by multiplication or division by phi.
If A + B = 12, then A = 12 / phi and B = 12 - A, which yields about A ≈ 7.416 and B ≈ 4.584. Entering A alone scales B = A / phi and the total = A + B. Entering B alone scales A = B × phi. Each path keeps the defining proportion.
Read the golden ratio formula
The constant phi equals (1 + √5) / 2, approximately 1.6180339887. The same number satisfies phi = 1 + 1/phi and phi^2 = phi + 1. QuickCalculators uses that exact algebraic definition rather than a short rounded stand-in when computing terms.
Plain text twin: phi equals open parenthesis 1 plus square root of 5 close parenthesis divided by 2. The golden mean and golden section are other names for the same ratio. Rounded displays of phi are for reading; internal math keeps higher precision.
Avoid this common mistake
Phi is irrational, so no ratio of whole numbers equals it exactly. Rounded golden-ratio terms no longer satisfy the proportion precisely. Calling a 5:8 or 8:13 pair "exact phi" is an approximation statement, not an identity. Fibonacci consecutive ratios approach phi but never equal it as exact rationals.
When the calculator prints rounded lengths, treat them as displays of irrational results. Exact work stays in terms of phi or √5.
Recognise the golden ratio identities
Useful identities include 1/phi = phi - 1 and phi^2 = phi + 1. Powers of phi reduce to linear combinations of phi and 1. Golden Ratio Calculator can surface those relations when explaining why scaling by phi and by 1/phi stay consistent.
Because 1/phi ≈ 0.618, the smaller part is about 61.8% of the larger part, and the larger part is about 61.8% of the whole. Those percentage readings are rounded views of the same irrational split.
Frequently asked questions
What is the golden ratio?
The golden ratio is the positive number phi such that a whole divided by the larger part equals the larger part divided by the smaller part. Both equalities equal phi. The calculator solves for those parts from any one known length.
What is the value of phi?
The value of phi is (1 + √5) / 2, about 1.6180339887. It is an irrational constant. Displays may round phi; the defining formula remains exact.
How do you calculate the golden ratio?
To calculate golden-ratio terms, set the known length and apply A = total / phi, B = total / phi^2, or A = B × phi depending on which value is given. The page runs those formulas once a single input is provided.
If A + B is 12, what are A and B?
If A + B is 12 in golden proportion, A is 12 / phi ≈ 7.416 and B is 12 - A ≈ 4.584. Those figures are rounded. Exact forms keep phi in the expression.
Why is phi irrational?
Phi is irrational because it equals (1 + √5) / 2 and √5 is irrational. No fraction of integers equals phi exactly. Approximations can be arbitrarily close without becoming exact.
What is the golden ratio used for?
The golden ratio is used in geometry, art, and design discussions as a named proportion, and in mathematics through Fibonacci limits and quadratic identities. This calculator focuses on computing the matching lengths, not on aesthetic advice.
Summary
The calculator fills in A, B, and A + B from any one known term using phi = (1 + √5) / 2. A whole of 12 splits into about 7.416 and 4.584 under that proportion. Phi is irrational, so whole-number ratios only approximate it. Identities such as phi^2 = phi + 1 keep the scaling relations consistent.