Length Of Golden Rectangle Given Diagonal Calculator

Golden Rectangle Length Formula:

\[ l = \frac{\phi}{\sqrt{1 + \phi^2}} \times d \]

Length of Golden Rectangle (l):

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1. What is the Golden Rectangle?

A Golden Rectangle is a rectangle whose side lengths are in the golden ratio, approximately 1:1.618. It's considered aesthetically pleasing and appears frequently in art, architecture, and nature.

2. How Does the Calculator Work?

The calculator uses the golden rectangle length formula:

\[ l = \frac{\phi}{\sqrt{1 + \phi^2}} \times d \]

Where:

\( l \) — Length of the golden rectangle
\( \phi \) — Golden ratio (≈1.618034)
\( d \) — Diagonal of the golden rectangle

Explanation: This formula derives from the Pythagorean theorem applied to a rectangle with sides in the golden ratio.

3. Mathematical Properties

Details: In a golden rectangle, the ratio of length to width equals the golden ratio φ. When a square is removed from a golden rectangle, the remaining rectangle is also a golden rectangle.

4. Using the Calculator

Tips: Enter the diagonal length of the golden rectangle in meters. The calculator will compute the corresponding length that maintains the golden ratio proportion.

5. Frequently Asked Questions (FAQ)

Q1: What is the exact value of the golden ratio?
A: The golden ratio φ = (1 + √5)/2 ≈ 1.6180339887498948482...

Q2: How is the width calculated from the length?
A: Width = Length / φ, since the length:width ratio equals the golden ratio.

Q3: Where are golden rectangles commonly found?
A: Golden rectangles appear in famous artworks, architectural designs (Parthenon), and even in credit card dimensions.

Q4: What's the relationship between the diagonal and sides?
A: The diagonal forms the hypotenuse of a right triangle with the length and width as legs, following d² = l² + w².

Q5: Can this calculator work with any units?
A: Yes, as long as you maintain consistent units. The result will be in the same units as your input.