Ridge Length Of Small Stellated Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{15 \times \sqrt{5+2\sqrt{5}}}{\frac{5}{4} \times (7+3\sqrt{5}) \times AV} \]

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 an important geometric measurement in this complex polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{15 \times \sqrt{5+2\sqrt{5}}}{\frac{5}{4} \times (7+3\sqrt{5}) \times AV} \]

Where:

\( l_{Ridge} \) — Ridge Length of Small Stellated Dodecahedron (meters)
\( AV \) — SA:V of Small Stellated Dodecahedron (1/meter)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the ridge length based on the surface area to volume ratio of the Small Stellated Dodecahedron, incorporating the golden ratio and geometric properties of this complex shape.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is important for understanding the geometric properties, structural integrity, and mathematical characteristics of the Small Stellated Dodecahedron in various applications including crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the surface area to volume ratio (SA:V) of the Small Stellated Dodecahedron in 1/meter units. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron, one of four regular nonconvex polyhedra. It consists of 12 pentagram faces with five meeting at each vertex.

Q2: What is the significance of the golden ratio in this formula?
A: The golden ratio \( \frac{1+\sqrt{5}}{2} \) appears naturally in the geometry of pentagrams and dodecahedrons, making it a fundamental component of the ridge length calculation.

Q3: What units should be used for the input?
A: The surface area to volume ratio should be entered in reciprocal meters (1/m), and the resulting ridge length will be calculated in meters.

Q4: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of positive values, but extremely small values close to zero may result in very large ridge lengths due to the inverse relationship in the formula.

Q5: What are typical values for SA:V of Small Stellated Dodecahedron?
A: The surface area to volume ratio depends on the specific dimensions of the polyhedron, but typical values range from 0.1 to 10 1/m for most practical applications.