Midsphere Radius Of Tetrakis Hexahedron Given Insphere Radius Calculator

Formula Used:

\[ r_m = \frac{1}{\sqrt{2}} \times \frac{10 \times r_i}{3 \times \sqrt{5}} \]

Midsphere Radius of Tetrakis Hexahedron:

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1. What is the Midsphere Radius of Tetrakis Hexahedron?

The Midsphere Radius of a Tetrakis Hexahedron is the radius of the sphere that is tangent to all the edges of the polyhedron. It represents the sphere that fits perfectly between the insphere and circumsphere of the Tetrakis Hexahedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_m = \frac{1}{\sqrt{2}} \times \frac{10 \times r_i}{3 \times \sqrt{5}} \]

Where:

\( r_m \) — Midsphere Radius of Tetrakis Hexahedron (m)
\( r_i \) — Insphere Radius of Tetrakis Hexahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{5} \) — Square root of 5 (approximately 2.2361)

Explanation: This formula establishes a precise mathematical relationship between the insphere radius and midsphere radius of a Tetrakis Hexahedron, derived from the geometric properties of this Catalan solid.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometric modeling, crystallography, and materials science where Tetrakis Hexahedron structures occur. It helps in understanding the spatial relationships and packing efficiency of such polyhedral structures.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and greater than zero. The calculator will automatically compute the corresponding midsphere radius using the established mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Tetrakis Hexahedron?
A: A Tetrakis Hexahedron is a Catalan solid that can be seen as a cube with square pyramids on each face. It has 24 faces (isosceles triangles), 36 edges, and 14 vertices.

Q2: How is the midsphere different from insphere and circumsphere?
A: The insphere is tangent to all faces, the circumsphere passes through all vertices, while the midsphere is tangent to all edges of the polyhedron.

Q3: What are the applications of Tetrakis Hexahedron?
A: This polyhedron appears in crystallography, architectural design, and as dice in some board games due to its interesting symmetry properties.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Tetrakis Hexahedron. Other polyhedra have different relationships between their insphere and midsphere radii.

Q5: What is the accuracy of this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Tetrakis Hexahedron. The result accuracy depends on the precision of the input value.