Insphere Radius of Pentakis Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \frac{\frac{76}{19}}{RA/V} \times \frac{\sqrt{413 + 162\sqrt{5}}}{23 + 11\sqrt{5}} \]

Insphere Radius of Pentakis Dodecahedron (r_i):

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1. What is Insphere Radius of Pentakis Dodecahedron?

The Insphere Radius of Pentakis Dodecahedron is the radius of the sphere that is contained by the Pentakis Dodecahedron in such a way that all the faces just touch the sphere. It represents the maximum sphere that can fit inside the polyhedron while touching all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \frac{\frac{76}{19}}{RA/V} \times \frac{\sqrt{413 + 162\sqrt{5}}}{23 + 11\sqrt{5}} \]

Where:

\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( RA/V \) — Surface to Volume Ratio of Pentakis Dodecahedron (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere 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 Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing efficiency, structural properties, and spatial relationships within the Pentakis Dodecahedron shape.

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 defined?
A: Surface to volume ratio is the ratio of the total surface area of a polyhedron to its total volume, measured in 1/m.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size and proportions of the specific Pentakis Dodecahedron, with smaller polyhedra having higher ratios.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Pentakis Dodecahedron only. Other polyhedra have different formulas for calculating insphere radius.

Q5: What are the practical applications of this calculation?
A: This calculation is used in crystallography, nanotechnology, architectural design, and any field where the geometric properties of Pentakis Dodecahedra are relevant.