Midsphere Radius of Deltoidal Hexecontahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_m = \frac{3}{20} \times (5 + 3\sqrt{5}) \times \frac{\frac{9}{45} \times \sqrt{10 \times (157 + 31\sqrt{5})}}{AV \times \frac{370 + 164\sqrt{5}}{25}} \]

Midsphere Radius of Deltoidal Hexecontahedron:

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1. What is Midsphere Radius of Deltoidal Hexecontahedron?

The Midsphere Radius of Deltoidal Hexecontahedron is the radius of the sphere for which all the edges of the Deltoidal Hexecontahedron become a tangent line on that sphere. It represents the sphere that touches the midpoint of each edge of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3}{20} \times (5 + 3\sqrt{5}) \times \frac{\frac{9}{45} \times \sqrt{10 \times (157 + 31\sqrt{5})}}{AV \times \frac{370 + 164\sqrt{5}}{25}} \]

Where:

\( r_m \) — Midsphere Radius of Deltoidal Hexecontahedron (m)
\( AV \) — SA:V of Deltoidal Hexecontahedron (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

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

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometric modeling, crystallography, and materials science where the deltoidal hexecontahedron structure appears. It helps in understanding the spatial relationships and packing efficiency of such structures.

4. Using the Calculator

Tips: Enter the surface area to volume ratio (SA:V) of the deltoidal hexecontahedron in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a deltoidal hexecontahedron?
A: A deltoidal hexecontahedron is a Catalan solid with 60 faces, each of which is a kite (deltoid). It is the dual polyhedron of the rhombicosidodecahedron.

Q2: What does SA:V ratio represent?
A: The surface area to volume ratio indicates how much surface area a structure has relative to its volume, which is important for understanding properties like diffusion and heat transfer.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the specific dimensions of the polyhedron. For a deltoidal hexecontahedron with unit edge length, the midsphere radius is approximately 2.3-2.8 units.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the deltoidal hexecontahedron. Other polyhedra have different formulas for calculating their midsphere radii.

Q5: What practical applications does this calculation have?
A: This calculation is used in fields such as crystallography, nanotechnology, and architectural design where complex geometric structures need to be analyzed and optimized.