Midsphere Radius Of Hexakis Octahedron Given Insphere Radius Calculator

Midsphere Radius of Hexakis Octahedron Formula:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \frac{2r_i}{\sqrt{\frac{402 + 195\sqrt{2}}{194}}} \]

Midsphere Radius of Hexakis Octahedron:

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1. What is the Midsphere Radius of Hexakis Octahedron?

The Midsphere Radius of 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:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \frac{2r_i}{\sqrt{\frac{402 + 195\sqrt{2}}{194}}} \]

Where:

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

Explanation: This formula calculates the midsphere radius based on the given insphere radius, using the geometric properties and relationships within the Hexakis Octahedron structure.

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 relationships and proportions within the Hexakis Octahedron structure, which has applications in crystallography, architecture, and mathematical research.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be a positive number greater than zero. The calculator will compute the corresponding midsphere radius using the mathematical relationship between these two geometric properties.

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: What is the difference between insphere and midsphere radius?
A: The insphere radius touches all faces of the polyhedron, while the midsphere radius touches all edges at their midpoints.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Octahedron due to its unique geometric properties.

Q4: What are the practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, 3D modeling, and mathematical research involving polyhedral structures.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Hexakis Octahedron, though practical measurements may have some margin of error.