1. What is the Insphere Radius of Hexakis Icosahedron?

The Insphere Radius of Hexakis Icosahedron is defined as the radius of the sphere that is contained by the Hexakis Icosahedron in such a way that all the faces just touch the sphere. It represents the maximum sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
• \( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
• \( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula establishes a precise mathematical relationship between the insphere radius and midsphere radius of a Hexakis Icosahedron, incorporating the golden ratio properties inherent in this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is crucial for understanding the geometric properties of Hexakis Icosahedrons, determining packing efficiency, and analyzing the polyhedron's internal structure in various mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the midsphere radius value in meters. The value must be positive and non-zero. The calculator will compute the corresponding insphere radius using the precise mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: How is the insphere radius different from the midsphere radius?
A: The insphere radius is the radius of the largest sphere that fits inside the polyhedron, while the midsphere radius is the radius of the sphere that touches all the edges of the polyhedron.

Q3: What are typical applications of this calculation?
A: This calculation is used in geometry research, architectural design, material science, and anywhere precise polyhedral measurements are required.

Q4: How accurate is this formula?
A: The formula is mathematically exact for perfect Hexakis Icosahedrons and provides precise results when implemented with sufficient numerical precision.

Q5: Can this calculator handle very large or very small values?
A: Yes, the calculator can process a wide range of positive values, though extremely large or small values may be limited by floating-point precision.