Midsphere Radius Of Hexakis Icosahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ r_m = \frac{1}{8} \times \frac{\frac{6}{5} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}{(RA/V \times \sqrt{6 \times (185 + 82 \times \sqrt{5})})} \times (5 + 3 \times \sqrt{5}) \]

Midsphere Radius (r_m):

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1. What is Midsphere Radius of Hexakis Icosahedron?

The Midsphere Radius of Hexakis Icosahedron is defined as the radius of the sphere for which all the edges of the Hexakis Icosahedron 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:

\[ r_m = \frac{1}{8} \times \frac{\frac{6}{5} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}{(RA/V \times \sqrt{6 \times (185 + 82 \times \sqrt{5})})} \times (5 + 3 \times \sqrt{5}) \]

Where:

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

Explanation: This formula calculates the midsphere radius based on the surface to volume ratio of a Hexakis Icosahedron, incorporating mathematical constants and relationships specific to this geometric shape.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of Hexakis Icosahedrons. It helps in determining the sphere that is tangent to all edges of the polyhedron, which has applications in crystallography, molecular modeling, and architectural design.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius of the Hexakis Icosahedron.

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. It has 120 faces, 180 edges, and 62 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 midsphere radius?
A: The midsphere radius depends on the specific dimensions of the Hexakis Icosahedron. For standard sizes, it typically ranges from centimeters to meters depending on the scale of the polyhedron.

Q4: Can this calculator be used for other polyhedrons?
A: No, this specific formula is designed only for Hexakis Icosahedrons. Other polyhedrons have different formulas for calculating midsphere radius.

Q5: What precision does this calculator provide?
A: The calculator provides results with up to 10 decimal places precision, which is sufficient for most geometric and engineering applications.