Short Edge of Hexakis Octahedron Given Insphere Radius Calculator

Formula Used:

\[ \text{Short Edge} = \left( \frac{2 \times r_i}{\sqrt{\frac{402 + 195\sqrt{2}}{194}}} \right) \times \sqrt{\frac{30 - 3\sqrt{2}}{60 + 6\sqrt{2}}} \]

Short Edge of Hexakis Octahedron:

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1. What is the Short Edge of Hexakis Octahedron?

The Short Edge of Hexakis Octahedron is the length of the shortest edge of any of the congruent triangular faces of the Hexakis Octahedron. It is an important geometric parameter in understanding the structure and properties of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \left( \frac{2 \times r_i}{\sqrt{\frac{402 + 195\sqrt{2}}{194}}} \right) \times \sqrt{\frac{30 - 3\sqrt{2}}{60 + 6\sqrt{2}}} \]

Where:

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

Explanation: This formula calculates the short edge length based on the insphere radius, incorporating the geometric relationships specific to the Hexakis Octahedron structure.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is crucial for understanding the geometric properties, surface area, volume, and other dimensional characteristics of the Hexakis Octahedron in mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding short edge length 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 cube. It has 48 congruent triangular faces, 72 edges, and 26 vertices.

Q2: What is the Insphere Radius?
A: The Insphere Radius is the radius of the sphere that is contained by the Hexakis Octahedron such that all faces just touch the sphere.

Q3: Are there other edges in a Hexakis Octahedron?
A: Yes, besides the short edge, a Hexakis Octahedron also has medium and long edges, creating a complex polyhedral structure.

Q4: What are typical applications of this calculation?
A: This calculation is used in geometry research, architectural design, crystal structure analysis, and mathematical modeling of complex polyhedra.

Q5: How accurate is this formula?
A: The formula is mathematically exact for a perfect Hexakis Octahedron and provides precise calculations when correct input values are used.