Insphere Radius of Hexakis Octahedron given Truncated Cuboctahedron Edge Calculator

Formula Used:

\[ r_i = \left( \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \right) \times \frac{2}{7} \times \sqrt{60 + 6\sqrt{2}} \times l_e \]

Insphere Radius (rᵢ):

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1. What is the Insphere Radius of Hexakis Octahedron?

The Insphere Radius of a Hexakis Octahedron is defined as the radius of the sphere that is contained by the Hexakis Octahedron in such a way that all the faces just touch the sphere. It represents the maximum 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:

\[ r_i = \left( \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \right) \times \frac{2}{7} \times \sqrt{60 + 6\sqrt{2}} \times l_e \]

Where:

\( r_i \) — Insphere Radius of Hexakis Octahedron (m)
\( l_e \) — Truncated Cuboctahedron Edge of Hexakis Octahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the insphere radius based on the truncated cuboctahedron edge length, incorporating mathematical constants and geometric relationships specific to the Hexakis Octahedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing properties, volume relationships, and spatial characteristics of polyhedral structures. It helps in determining how spheres or other objects can be arranged within the polyhedron.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron. It has 48 faces, 72 edges, and 26 vertices.

Q2: What is a Truncated Cuboctahedron Edge?
A: The truncated cuboctahedron edge refers to the edge length of the polyhedron created by truncating the vertices of a cuboctahedron, which is used to construct the Hexakis Octahedron.

Q3: What are typical values for the Insphere Radius?
A: The insphere radius depends on the edge length of the truncated cuboctahedron. For typical geometric applications, values can range from millimeters to meters depending on the scale of the polyhedron.

Q4: Are there limitations to this calculation?
A: This formula is specifically designed for regular Hexakis Octahedrons. It may not be accurate for irregular or modified polyhedral structures.

Q5: What practical applications does this calculation have?
A: This calculation is used in crystallography, architectural design, 3D modeling, and materials science where precise geometric relationships of polyhedral structures are important.