Pentagram Chord of Small Stellated Dodecahedron Given Ridge Length Calculator

Formula Used:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \frac{2 \times lRidge}{1+\sqrt{5}} \]

Pentagram Chord of Small Stellated Dodecahedron:

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1. What is the Pentagram Chord of Small Stellated Dodecahedron?

The Pentagram Chord of Small Stellated Dodecahedron is the distance between any pair of non-adjacent peak vertices of the pentagram corresponding to the Small Stellated Dodecahedron. It represents a key geometric measurement in this complex polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \frac{2 \times lRidge}{1+\sqrt{5}} \]

Where:

\( lc(Pentagram) \) — Pentagram Chord of Small Stellated Dodecahedron (m)
\( lRidge \) — Ridge Length of Small Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the precise mathematical relationship between the ridge length and the pentagram chord in a Small Stellated Dodecahedron, incorporating the golden ratio properties inherent in this geometric structure.

3. Importance of Pentagram Chord Calculation

Details: Accurate calculation of the pentagram chord is essential for geometric analysis, architectural design applications, and understanding the mathematical properties of stellated polyhedra. It helps in determining the complete dimensional relationships within the Small Stellated Dodecahedron.

4. Using the Calculator

Tips: Enter the ridge length in meters. The value must be positive and valid. The calculator will compute the corresponding pentagram chord using the established mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron with pentagrammic faces.

Q2: How is the ridge length defined?
A: The ridge length is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Small Stellated Dodecahedron.

Q3: What practical applications does this calculation have?
A: This calculation is used in mathematical geometry, architectural design, crystal structure analysis, and in creating accurate models of complex polyhedral structures.

Q4: Why does the formula include √5?
A: The square root of 5 appears because the Small Stellated Dodecahedron incorporates golden ratio relationships, and √5 is fundamental to golden ratio mathematics.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron. Other polyhedra have different geometric relationships and require different formulas.