Midsphere Radius of Hexakis Octahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \frac{12\sqrt{543 + 176\sqrt{2}}}{RA/V \times \sqrt{6(986 + 607\sqrt{2})}} \]

Midsphere Radius (rₘ):

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1. What is the Midsphere Radius of Hexakis Octahedron?

The Midsphere Radius of a Hexakis Octahedron is defined as the radius of the sphere for which all the edges of the Hexakis Octahedron become a tangent line on that sphere. It represents the sphere that touches the midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \frac{12\sqrt{543 + 176\sqrt{2}}}{RA/V \times \sqrt{6(986 + 607\sqrt{2})}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the midsphere radius based on the surface to volume ratio of the Hexakis Octahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometric analysis and 3D modeling of polyhedra. It helps in understanding the spatial properties and proportions of the Hexakis Octahedron, which has applications in crystallography, architecture, and mathematical research.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius 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 faces, 72 edges, and 26 vertices.

Q2: How is the surface to volume ratio related to the midsphere radius?
A: The surface to volume ratio and midsphere radius are inversely related through a complex mathematical relationship specific to the geometry of the Hexakis Octahedron.

Q3: What are typical values for the midsphere radius?
A: The midsphere radius depends on the specific dimensions of the Hexakis Octahedron. For a given surface to volume ratio, the radius can be calculated using this formula.

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

Q5: What are the practical applications of this calculation?
A: This calculation is used in geometric modeling, crystallography, architectural design, and mathematical research involving polyhedral structures.