Ridge Length of Great Dodecahedron given Pyramidal Height Calculator

Formula Used:

\[ l_{Ridge} = \frac{\sqrt{5}-1}{2} \times \frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3-\sqrt{5})} \]

Ridge Length of Great Dodecahedron:

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1. What is the Ridge Length of Great Dodecahedron?

The Ridge Length of Great Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Dodecahedron. It is an important geometric measurement in the study of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{\sqrt{5}-1}{2} \times \frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3-\sqrt{5})} \]

Where:

\( l_{Ridge} \) — Ridge Length of Great Dodecahedron (m)
\( h_{Pyramid} \) — Pyramidal Height of Great Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the ridge length based on the pyramidal height using the mathematical relationships inherent in the geometry of the great dodecahedron.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for understanding the geometric properties of the great dodecahedron, which has applications in various fields including crystallography, architecture, and mathematical modeling of complex structures.

4. Using the Calculator

Tips: Enter the pyramidal height in meters. The value must be positive and greater than zero. The calculator will compute the corresponding ridge length of the great dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A great dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagonal faces. It is one of four regular star polyhedra.

Q2: How is the Ridge Length measured?
A: The ridge length is measured as the straight-line distance between an inwards directed pyramidal apex and any of its adjacent peak vertices.

Q3: What units should be used for input?
A: The calculator accepts input in meters, but any consistent unit of length can be used as long as the same unit is used throughout the calculation.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric great dodecahedron and may not account for manufacturing tolerances or material properties in physical implementations.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the great dodecahedron and may not apply to other polyhedral shapes.