Insphere Radius of Hexakis Octahedron Calculator

Formula Used:

\[ r_i = \frac{l_{long}}{2} \times \sqrt{\frac{402 + 195\sqrt{2}}{194}} \]

Insphere Radius of Hexakis Octahedron:

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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:

\[ r_i = \frac{l_{long}}{2} \times \sqrt{\frac{402 + 195\sqrt{2}}{194}} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Octahedron (m)
\( l_{long} \) — Long Edge of Hexakis Octahedron (m)

Explanation: This formula calculates the radius of the inscribed sphere based on the long edge length of the Hexakis Octahedron, using a specific mathematical relationship derived from the geometry of this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the spatial properties of polyhedra, determining packing efficiency, and analyzing the internal volume characteristics of complex geometric shapes.

4. Using the Calculator

Tips: Enter the long edge length of the Hexakis Octahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.

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: How is the insphere radius different from the circumsphere radius?
A: The insphere radius is the radius of the largest sphere that fits inside the polyhedron and touches all faces, while the circumsphere radius is the radius of the smallest sphere that contains the polyhedron and touches all vertices.

Q3: What are the practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and in the study of polyhedral structures in various scientific and engineering fields.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Hexakis Octahedron only. Other polyhedra have different formulas for calculating their insphere radii.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect Hexakis Octahedron. The accuracy depends on the precision of the input value and the computational precision of the calculator.