Insphere Radius of Hexakis Octahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \frac{12\sqrt{543 + 176\sqrt{2}}}{R_{A/V} \times \sqrt{6(986 + 607\sqrt{2})}} \]

Insphere Radius (r_i):

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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 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:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \frac{12\sqrt{543 + 176\sqrt{2}}}{R_{A/V} \times \sqrt{6(986 + 607\sqrt{2})}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( R_{A/V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: The formula calculates the insphere radius based on the surface to volume ratio of the Hexakis Octahedron, incorporating various mathematical constants and square root operations.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing efficiency, structural properties, and spatial relationships within polyhedral structures like the Hexakis Octahedron.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and 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 surface to volume ratio defined?
A: Surface to volume ratio is the ratio of the total surface area of a polyhedron to its total volume, measured in m⁻¹.

Q3: What are typical values for insphere radius?
A: The insphere radius varies depending on the size and proportions of the Hexakis Octahedron, typically ranging from fractions of a meter to several meters.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect Hexakis Octahedron shape and may not account for manufacturing tolerances or material deformations.

Q5: What practical applications does this have?
A: This calculation is useful in crystallography, nanotechnology, architectural design, and any field dealing with polyhedral structures and their geometric properties.