Ridge Length of Small Stellated Dodecahedron Calculator

Formula Used:

\[ l_{Ridge} = \frac{1 + \sqrt{5}}{2} \times l_e \]

Ridge Length of Small Stellated Dodecahedron:

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

The Ridge Length of Small Stellated Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Small Stellated Dodecahedron. It is a key geometric measurement in this polyhedron's structure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{1 + \sqrt{5}}{2} \times l_e \]

Where:

\( l_{Ridge} \) — Ridge Length of Small Stellated Dodecahedron (m)
\( l_e \) — Edge Length of Small Stellated Dodecahedron (m)
\( \frac{1 + \sqrt{5}}{2} \) — Golden ratio constant (approximately 1.618)

Explanation: The formula demonstrates that the ridge length is proportional to the edge length through the golden ratio, which is a fundamental mathematical constant often found in geometric relationships.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for understanding the geometric properties of the Small Stellated Dodecahedron, including its symmetry, proportions, and spatial relationships between different elements of the polyhedron.

4. Using the Calculator

Tips: Enter the edge length of the Small Stellated Dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding ridge length using the golden ratio relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is the Small Stellated Dodecahedron?
A: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron, one of four regular star polyhedra. It is formed by stellating a regular dodecahedron.

Q2: Why does the formula use the golden ratio?
A: The golden ratio appears naturally in the geometry of regular dodecahedra and their stellations due to the mathematical properties of pentagonal symmetry.

Q3: What are the applications of this calculation?
A: This calculation is used in mathematical geometry, architectural design, crystalography studies, and in understanding the properties of regular star polyhedra.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron. Other polyhedra have different geometric relationships and formulas.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact when using the golden ratio constant. The accuracy depends on the precision of the input edge length value.