Insphere Radius of Deltoidal Hexecontahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \frac{\frac{9}{45} \cdot \sqrt{10(157 + 31\sqrt{5})}}{SA:V \cdot \frac{370 + 164\sqrt{5}}{25}} \]

Insphere Radius of Deltoidal Hexecontahedron:

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1. What is 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 this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \frac{\frac{9}{45} \cdot \sqrt{10(157 + 31\sqrt{5})}}{SA:V \cdot \frac{370 + 164\sqrt{5}}{25}} \]

Where:

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

Explanation: This formula calculates the insphere radius based on the surface to volume ratio of the deltoidal hexecontahedron, incorporating mathematical constants related to this specific polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important for understanding the geometric properties of the deltoidal hexecontahedron, particularly in applications involving packing problems, material science, and geometric modeling where internal sphere fitting is relevant.

4. Using the Calculator

Tips: Enter the surface 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 deltoid (kite-shaped) faces, 120 edges, and 62 vertices.

Q2: How is Surface to Volume Ratio defined?
A: Surface to Volume Ratio (SA:V) is the total surface area of a polyhedron divided by its total volume.

Q3: What are typical values for SA:V of Deltoidal Hexecontahedron?
A: The SA:V depends on the specific dimensions of the polyhedron but generally ranges based on the size and proportions of the deltoidal hexecontahedron.

Q4: Why does the formula contain so many square roots and constants?
A: The constants and square roots come from the mathematical properties of the golden ratio (φ) and the specific geometry of the deltoidal hexecontahedron, which is related to the icosahedral symmetry group.

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