Circumsphere Radius of Small Stellated Dodecahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{50 + 22\sqrt{5}}}{4} \times \frac{15\sqrt{5 + 2\sqrt{5}}}{\frac{5}{4}(7 + 3\sqrt{5}) \times AV} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Small Stellated Dodecahedron?

The circumsphere radius of a small stellated dodecahedron is the radius of the sphere that completely contains the polyhedron, with all vertices lying on the sphere's surface. It's a fundamental geometric property of this Kepler-Poinsot solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{50 + 22\sqrt{5}}}{4} \times \frac{15\sqrt{5 + 2\sqrt{5}}}{\frac{5}{4}(7 + 3\sqrt{5}) \times AV} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( AV \) — Surface to volume ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the circumsphere radius based on the surface to volume ratio of the small stellated dodecahedron, incorporating the mathematical constants specific to this polyhedron's geometry.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, crystallography, and materials science for understanding the spatial properties and packing efficiency of this complex polyhedral form.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a small stellated dodecahedron?
A: It's one of the four Kepler-Poinsot solids, formed by extending the faces of a regular dodecahedron until they intersect.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the small stellated dodecahedron, which involves the golden ratio and its mathematical properties.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the polyhedron, with smaller specimens having higher ratios.

Q4: Can this calculator be used for other polyhedra?
A: No, this formula is specific to the small stellated dodecahedron due to its unique geometric properties.

Q5: What practical applications does this calculation have?
A: It's used in mathematical research, architectural design, and in understanding the geometric properties of complex natural and synthetic structures.