Midsphere Radius of Pentakis Dodecahedron given Surface to Volume Ratio Calculator

Midsphere Radius of Pentakis Dodecahedron Formula:

\[ r_m = \frac{3+\sqrt{5}}{4} \times \frac{76/19}{RA/V} \times \frac{\sqrt{413+162\sqrt{5}}}{23+11\sqrt{5}} \]

Midsphere Radius (rₘ):

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1. What is Midsphere Radius of Pentakis Dodecahedron?

The Midsphere Radius of Pentakis Dodecahedron is the radius of the sphere for which all the edges of the Pentakis Dodecahedron become a tangent line on that sphere. It represents the sphere that touches the midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3+\sqrt{5}}{4} \times \frac{76/19}{RA/V} \times \frac{\sqrt{413+162\sqrt{5}}}{23+11\sqrt{5}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the midsphere radius based on the surface to volume ratio of the Pentakis Dodecahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and crystallography for understanding the spatial properties and symmetry of the Pentakis Dodecahedron. It helps in determining the optimal sphere that fits within the polyhedron's structure.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 60 faces, 90 edges, and 32 vertices.

Q2: How is surface to volume ratio related to midsphere radius?
A: The surface to volume ratio affects the midsphere radius through an inverse relationship - as surface to volume ratio increases, the midsphere radius decreases, and vice versa.

Q3: What are typical values for midsphere radius?
A: The midsphere radius values depend on the specific dimensions of the Pentakis Dodecahedron and can vary significantly based on the surface to volume ratio.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Pentakis Dodecahedron. Other polyhedra have different geometric relationships and formulas.

Q5: What practical applications does this calculation have?
A: This calculation is used in crystallography, material science, and geometric modeling where Pentakis Dodecahedron structures appear, such as in certain virus structures and fullerene molecules.