Edge Length of Small Stellated Dodecahedron given Ridge Length Calculator

Formula Used:

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

Edge Length of Small Stellated Dodecahedron:

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

The Edge Length of Small Stellated Dodecahedron given Ridge Length is a geometric calculation that determines the length of the edges of a Small Stellated Dodecahedron based on its ridge length measurement.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( l_e \) — Edge Length of Small Stellated Dodecahedron (m)
\( l_{Ridge} \) — Ridge Length of Small Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the ridge length and edge length of a Small Stellated Dodecahedron using the golden ratio properties.

3. Importance of Edge Length Calculation

Details: Accurate edge length calculation is crucial for geometric modeling, architectural design, and mathematical analysis of polyhedral structures, particularly in understanding the properties of stellated dodecahedrons.

4. Using the Calculator

Tips: Enter the ridge length in meters. The value must be positive and valid. The calculator will compute the corresponding edge length using the mathematical relationship between these two geometric parameters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: A Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron that represents one of the four regular star polyhedra, formed by extending the faces of a regular dodecahedron.

Q2: What is the relationship between edge length and ridge length?
A: The edge length is derived from the ridge length through a mathematical formula involving the golden ratio, specifically \( \frac{1 + \sqrt{5}}{2} \).

Q3: What units should be used for input?
A: The calculator uses meters as the default unit, but the formula works with any consistent unit system (the result will be in the same units as the input).

Q4: Are there any limitations to this calculation?
A: This formula is specifically designed for the geometric properties of a perfect Small Stellated Dodecahedron and assumes ideal mathematical conditions.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula and calculator are designed exclusively for the Small Stellated Dodecahedron and its geometric properties.