Midsphere Radius of Hexakis Icosahedron given Truncated Icosidodecahedron Edge Calculator

Formula Used:

\[ r_m = \frac{5 + 3\sqrt{5}}{8} \times \frac{2}{5} \times l_e \times \sqrt{15(5 - \sqrt{5})} \]

Midsphere Radius of Hexakis Icosahedron:

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1. What is the Midsphere Radius of Hexakis Icosahedron?

The Midsphere Radius of a 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 represents the sphere that touches all the edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_m = \frac{5 + 3\sqrt{5}}{8} \times \frac{2}{5} \times l_e \times \sqrt{15(5 - \sqrt{5})} \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
\( l_e \) — Truncated Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the truncated edge length of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is crucial for understanding the geometric properties of the Hexakis Icosahedron, its spatial relationships, and for applications in crystallography, molecular modeling, and architectural design.

4. Using the Calculator

Tips: Enter the truncated edge length of the Hexakis Icosahedron in meters. The value must be positive and greater than zero for accurate calculation.

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 icosahedron, featuring 120 faces, 180 edges, and 62 vertices.

Q2: How is the truncated edge defined?
A: The truncated edge refers to the length of the edges created by truncating the vertices of an Icosidodecahedron to form the Hexakis Icosahedron.

Q3: What are typical values for midsphere radius?
A: The midsphere radius varies depending on the size of the polyhedron, but it's always proportional to the edge length with the given mathematical relationship.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Icosahedron and its relationship between truncated edge length and midsphere radius.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most geometric and engineering applications.