Midsphere Radius of Hexakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ r_m = \frac{1}{8} \times \left( \frac{4 \times r_i}{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}} \right) \times (5 + 3 \times \sqrt{5}) \]

Midsphere Radius of Hexakis Icosahedron:

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1. What is the Midsphere Radius of Hexakis Icosahedron?

The Midsphere Radius of Hexakis Icosahedron is defined as the radius of the sphere for which all the edges of the Hexakis Icosahedron become a tangent line on that sphere. It is an important geometric property of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1}{8} \times \left( \frac{4 \times r_i}{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}} \right) \times (5 + 3 \times \sqrt{5}) \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the midsphere radius and insphere radius of a Hexakis Icosahedron, incorporating the mathematical constant √5 which is fundamental to this geometric shape.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is crucial for understanding the geometric properties of Hexakis Icosahedron, particularly in fields of geometry, crystallography, and materials science where this shape appears.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius using the established 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 midsphere different from the insphere?
A: The insphere is tangent to all faces, while the midsphere is tangent to all edges of the polyhedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in geometric modeling, crystallography, and the study of complex polyhedral structures in materials science.

Q4: Why does the formula contain √5?
A: The √5 appears because the Hexakis Icosahedron is related to the icosahedron, which has golden ratio proportions involving √5.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Hexakis Icosahedron, assuming precise input values.