Insphere Radius of Hexakis Octahedron given Medium Edge Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \frac{14 \times l_{medium}}{3 \times (1 + 2\sqrt{2})} \]

Insphere Radius of Hexakis Octahedron:

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1. What is the Insphere Radius of Hexakis Octahedron?

The Insphere Radius of 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 largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \frac{14 \times l_{medium}}{3 \times (1 + 2\sqrt{2})} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Octahedron (m)
\( l_{medium} \) — Medium Edge of Hexakis Octahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the insphere radius based on the medium edge length of the Hexakis Octahedron, using mathematical constants and geometric relationships specific to this polyhedron.

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 the Hexakis Octahedron shape.

4. Using the Calculator

Tips: Enter the medium edge length of the Hexakis Octahedron in meters. The value must be positive and valid for accurate calculation.

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 medium edge defined?
A: The medium edge of Hexakis Octahedron is the length of the medium edge of any of the congruent triangular faces of the polyhedron.

Q3: What are typical values for the insphere radius?
A: The insphere radius depends on the size of the polyhedron. For a unit medium edge length, the insphere radius is approximately 0.75 units.

Q4: Are there limitations to this formula?
A: This formula is specifically derived for the Hexakis Octahedron and assumes perfect geometric proportions. It may not apply to distorted or irregular variations.

Q5: What practical applications does this calculation have?
A: This calculation is useful in crystallography, molecular modeling, and architectural design where Hexakis Octahedron shapes occur.