Insphere Radius of Deltoidal Hexecontahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \left( \frac{11V}{45 \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}}} \right)^{\frac{1}{3}} \]

Insphere Radius of Deltoidal Hexecontahedron:

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1. What is the Insphere Radius of Deltoidal Hexecontahedron?

The Insphere Radius of Deltoidal Hexecontahedron is the radius of the sphere that is contained by the Deltoidal Hexecontahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the polyhedron while being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \left( \frac{11V}{45 \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}}} \right)^{\frac{1}{3}} \]

Where:

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

Explanation: This formula calculates the insphere radius based on the volume of the deltoidal hexecontahedron, using mathematical constants derived from the geometric properties of this specific polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing properties, structural integrity, and spatial relationships within polyhedral structures.

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 (kite-shaped) faces, 62 vertices, and 120 edges.

Q2: Why is this formula so complex?
A: The formula incorporates mathematical constants derived from the specific geometric properties and symmetry of the deltoidal hexecontahedron, resulting in a complex but precise calculation.

Q3: What are typical values for the insphere radius?
A: The insphere radius depends on the volume of the polyhedron. For a given volume, the insphere radius will be smaller than the circumsphere radius but larger than the midsphere radius.

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

Q5: What practical applications does this calculation have?
A: This calculation is used in crystallography, molecular modeling, architectural design, and any field that involves the study of polyhedral structures and their geometric properties.