Midsphere Radius Of Tetrakis Hexahedron Given Volume Calculator

Formula Used:

\[ r_m = \frac{1}{\sqrt{2}} \times \left( \frac{2 \times V}{3} \right)^{1/3} \]

Midsphere Radius of Tetrakis Hexahedron (rₘ):

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1. What is the Midsphere Radius of Tetrakis Hexahedron?

The Midsphere Radius of Tetrakis Hexahedron is the radius of the sphere for which all the edges of the Tetrakis Hexahedron 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:

\[ r_m = \frac{1}{\sqrt{2}} \times \left( \frac{2 \times V}{3} \right)^{1/3} \]

Where:

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

Explanation: This formula calculates the midsphere radius based on the volume of the Tetrakis Hexahedron, using the mathematical relationship between volume and radius in this specific polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of the Tetrakis Hexahedron. It helps in various applications including material science, crystallography, and architectural design where this polyhedral form is used.

4. Using the Calculator

Tips: Enter the volume of the Tetrakis Hexahedron in cubic meters. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius.

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, 36 edges, and 14 vertices.

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

Q3: What are typical applications of this calculation?
A: This calculation is used in geometric modeling, crystal structure analysis, and in fields where precise spatial relationships of polyhedra are important.

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 volume and midsphere radius.

Q5: What units should I use for the volume input?
A: The calculator expects volume in cubic meters (m³), but you can use any consistent unit system as long as the result is interpreted in the same units.