Insphere Radius of Hexakis Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \frac{6/5}{RA/V} \cdot \sqrt{\frac{10 \cdot (417 + 107\sqrt{5})}{6 \cdot (185 + 82\sqrt{5})}} \]

Insphere Radius (r_i):

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1. What is Insphere Radius of Hexakis Icosahedron?

The Insphere Radius of a Hexakis Icosahedron is defined as the radius of the sphere that is contained by the Hexakis Icosahedron in such a way that all the faces just touch the sphere. It represents the maximum sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \frac{6/5}{RA/V} \cdot \sqrt{\frac{10 \cdot (417 + 107\sqrt{5})}{6 \cdot (185 + 82\sqrt{5})}} \]

Where:

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

Explanation: The formula calculates the insphere radius based on the surface to volume ratio of the Hexakis Icosahedron, 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 complex polyhedral structures like the Hexakis Icosahedron.

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 Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron, featuring 120 faces, 180 edges, and 62 vertices.

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is crucial in determining various physical and chemical properties of materials, including reaction rates, heat transfer, and mechanical strength.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size and shape of the polyhedron, with smaller structures typically having higher ratios.

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

Q5: What units should I use for the calculation?
A: Use consistent units - meters for length and 1/meters for surface to volume ratio to get the insphere radius in meters.