Insphere Radius of Tetrakis Hexahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{3\sqrt{5}}{10} \times \left(\frac{2V}{3}\right)^{\frac{1}{3}} \]

Insphere Radius of Tetrakis Hexahedron:

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1. What is the Insphere Radius of Tetrakis Hexahedron?

The Insphere Radius of Tetrakis Hexahedron is the radius of the sphere that is contained by the Tetrakis Hexahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3\sqrt{5}}{10} \times \left(\frac{2V}{3}\right)^{\frac{1}{3}} \]

Where:

\( r_i \) — Insphere Radius of Tetrakis Hexahedron (m)
\( V \) — Volume of Tetrakis Hexahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the insphere radius based on the volume of the Tetrakis Hexahedron, using mathematical constants and cube root operations.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and 3D modeling as it helps determine the maximum size of a sphere that can be inscribed within the polyhedron, which has applications in packaging, material science, and architectural design.

4. Using the Calculator

Tips: Enter the volume of the Tetrakis Hexahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Tetrakis Hexahedron?
A: A Tetrakis Hexahedron is a Catalan solid that is the dual of the truncated octahedron. It has 24 isosceles triangular faces, 14 vertices, and 36 edges.

Q2: How is the volume related to the insphere radius?
A: The volume and insphere radius have a cubic relationship, meaning as the volume increases, the insphere radius increases proportionally to the cube root of the volume.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is only applicable to the Tetrakis Hexahedron. Other polyhedra have different formulas for calculating their insphere radii.

Q4: What are the practical applications of this calculation?
A: This calculation is useful in materials science, crystallography, and 3D modeling where understanding the internal geometry of polyhedra is important.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect Tetrakis Hexahedron. In practical applications, accuracy depends on the precision of the volume measurement.