Truncated Icosidodecahedron Edge of Hexakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ \text{Truncated Edge} = \frac{5}{2\sqrt{15(5-\sqrt{5})}} \times \frac{4r_i}{\sqrt{\frac{15}{241}(275+119\sqrt{5})}} \]

Truncated Edge of Hexakis Icosahedron:

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1. What is Truncated Icosidodecahedron Edge of Hexakis Icosahedron?

The Truncated Edge of Hexakis Icosahedron is the length of the edges of a Hexakis Icosahedron that is created by truncating the vertices of an Icosidodecahedron. This geometric measurement is important in understanding the properties of these complex polyhedra.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Truncated Edge} = \frac{5}{2\sqrt{15(5-\sqrt{5})}} \times \frac{4r_i}{\sqrt{\frac{15}{241}(275+119\sqrt{5})}} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the truncated edge length based on the insphere radius using geometric relationships derived from the properties of the Hexakis Icosahedron.

3. Importance of This Calculation

Details: Calculating the truncated edge length is essential for understanding the geometric properties of Hexakis Icosahedrons, which have applications in various fields including crystallography, architecture, and mathematical modeling of complex structures.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding truncated edge length.

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. It has 120 faces, 180 edges, and 62 vertices.

Q2: What is the relationship between insphere radius and truncated edge?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, touching all faces. The truncated edge length is derived from this measurement through geometric relationships.

Q3: What are typical values for these measurements?
A: The actual values depend on the specific polyhedron size, but the ratio between insphere radius and truncated edge remains constant for all Hexakis Icosahedrons.

Q4: Are there practical applications of this calculation?
A: Yes, these calculations are used in various fields including molecular modeling, architectural design, and the study of geometric structures in mathematics.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, assuming precise input values and proper implementation of the formula.