Midsphere Radius of Deltoidal Hexecontahedron given Volume Calculator

Formula Used:

\[ r_m = \frac{3}{20} \times (5 + 3\sqrt{5}) \times \left( \frac{11V}{45\sqrt{\frac{370 + 164\sqrt{5}}{25}}} \right)^{\frac{1}{3}} \]

Midsphere Radius of Deltoidal Hexecontahedron:

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1. What is the 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 midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3}{20} \times (5 + 3\sqrt{5}) \times \left( \frac{11V}{45\sqrt{\frac{370 + 164\sqrt{5}}{25}}} \right)^{\frac{1}{3}} \]

Where:

\( r_m \) — Midsphere Radius of Deltoidal Hexecontahedron (m)
\( V \) — Volume of Deltoidal Hexecontahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the volume 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 geometry and crystallography for understanding the spatial properties and symmetry of the deltoidal hexecontahedron, which has applications in material science and mathematical modeling.

4. Using the Calculator

Tips: Enter the volume of the deltoidal hexecontahedron in cubic meters. 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 deltoid faces, 120 edges, and 62 vertices. It is the dual polyhedron of the rhombicosidodecahedron.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the deltoidal hexecontahedron, which involves irrational numbers and specific mathematical relationships that describe its unique properties.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, and mathematical research involving polyhedral structures and their geometric properties.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact when using precise values for the constants. The accuracy in practice depends on the precision of the input volume value.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal hexecontahedron. Other polyhedra have different formulas for calculating their midsphere radii.