Insphere Radius of Hexakis Octahedron Given Volume Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \left( \frac{28V}{\sqrt{6(986 + 607\sqrt{2})}} \right)^{\frac{1}{3}} \]

Insphere Radius (rᵢ):

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1. What is 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 largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \left( \frac{28V}{\sqrt{6(986 + 607\sqrt{2})}} \right)^{\frac{1}{3}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( V \) — Volume of Hexakis Octahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the insphere radius based on the volume of the Hexakis Octahedron, using specific mathematical constants derived from the geometry of this particular 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 structure of crystalline materials.

4. Using the Calculator

Tips: Enter the volume of the Hexakis Octahedron in cubic meters. The volume must be a positive value greater than zero 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 this formula derived?
A: The formula is derived from the geometric properties of the Hexakis Octahedron, specifically the relationship between its volume and the radius of its inscribed sphere.

Q3: What are typical values for insphere radius?
A: The insphere radius varies depending on the volume of the polyhedron. For a given volume, the insphere radius can be calculated using this specific formula.

Q4: Can this calculator handle very large or very small volumes?
A: Yes, the calculator can handle a wide range of volume values, as long as they are positive numbers.

Q5: What units should I use for volume input?
A: The calculator expects volume input in cubic meters (m³). If your volume is in different units, convert it to cubic meters first.