Insphere Radius of Deltoidal Hexecontahedron given Total Surface Area Calculator

Formula Used:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \sqrt{\frac{11 \cdot TSA}{9 \cdot \sqrt{10 \cdot (157 + 31\sqrt{5})}}} \]

Insphere Radius (rᵢ):

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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 the polyhedron while being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \sqrt{\frac{11 \cdot TSA}{9 \cdot \sqrt{10 \cdot (157 + 31\sqrt{5})}}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( TSA \) — Total Surface Area of Deltoidal Hexecontahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the total surface area of the deltoidal hexecontahedron, using mathematical constants and relationships specific to this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the spatial properties of polyhedra, packing efficiency, and various engineering applications involving polyhedral structures.

4. Using the Calculator

Tips: Enter the total surface area of the deltoidal hexecontahedron in square 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 polyhedron with 60 deltoid (kite-shaped) faces. It is a Catalan solid and the dual of the rhombicosidodecahedron.

Q2: What units should I use for the input?
A: The calculator expects the total surface area in square meters (m²). Make sure to use consistent units throughout your calculations.

Q3: Can this calculator handle very large or very small values?
A: Yes, the calculator can handle a wide range of values, but extremely large or small numbers may affect computational precision.

Q4: What is the typical range for insphere radius values?
A: The insphere radius depends on the size of the polyhedron. For a deltoidal hexecontahedron with unit surface area, the insphere radius is approximately 0.2-0.3 units.

Q5: Are there any limitations to this calculation?
A: The formula assumes a perfect deltoidal hexecontahedron shape. Real-world applications may require adjustments for manufacturing tolerances or material properties.