Midsphere Radius Of Hexakis Octahedron Given Volume Calculator

Formula Used:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \left( \frac{28V}{\sqrt{6(986 + 607\sqrt{2})}} \right)^{\frac{1}{3}} \]

Midsphere Radius of Hexakis Octahedron (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 midpoint of every edge of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \left( \frac{28V}{\sqrt{6(986 + 607\sqrt{2})}} \right)^{\frac{1}{3}} \]

Where:

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

Explanation: The formula calculates the midsphere radius based on the volume 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 modeling, crystallography, and materials science where Hexakis Octahedrons appear. It helps in understanding the spatial relationships and packing efficiency of such structures.

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 the midsphere radius different from the insphere radius?
A: The midsphere radius touches the midpoints of all edges, while the insphere radius is tangent to all faces of the polyhedron.

Q3: What are typical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, and the study of geometric properties of complex polyhedra.

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

Q5: What units should be used for input?
A: The calculator expects volume input in cubic meters, and returns the midsphere radius in meters. Consistent units must be maintained throughout.