Ridge Length of Great Stellated Dodecahedron Given Volume Calculator

Formula Used:

\[ l_{Ridge} = \frac{1 + \sqrt{5}}{2} \times \left( \frac{4 \times V}{5 \times (3 + \sqrt{5})} \right)^{\frac{1}{3}} \]

Ridge Length of Great Stellated Dodecahedron:

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1. What is the Ridge Length of Great Stellated Dodecahedron?

The Ridge Length of Great Stellated Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Stellated Dodecahedron. It is a key geometric parameter in understanding the structure of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{1 + \sqrt{5}}{2} \times \left( \frac{4 \times V}{5 \times (3 + \sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

\( l_{Ridge} \) — Ridge Length of Great Stellated Dodecahedron (m)
\( V \) — Volume of Great Stellated Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives the ridge length from the volume of the polyhedron using the golden ratio and cubic root relationship.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for geometric analysis, architectural design applications, and understanding the spatial properties of the Great Stellated Dodecahedron in mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the volume of the Great Stellated Dodecahedron in cubic meters. The volume must be a positive value greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the Great Stellated Dodecahedron?
A: The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra, formed by extending the faces of a regular dodecahedron until they intersect.

Q2: Why is the golden ratio (φ) present in the formula?
A: The golden ratio appears naturally in the geometry of regular dodecahedra and their stellated forms due to their pentagonal symmetry.

Q3: What are typical volume values for this polyhedron?
A: Volume values depend on the scale of the polyhedron. For a unit edge length, the volume is approximately 7.66 cubic units.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Stellated Dodecahedron due to its unique geometric properties.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect Great Stellated Dodecahedron, though real-world measurements may introduce practical limitations.