Insphere Radius of Tetrakis Hexahedron given Pyramidal Edge Length Calculator

Formula Used:

\[ r_i = \frac{2 \times l_e(Pyramid)}{\sqrt{5}} \]

Insphere Radius of Tetrakis Hexahedron:

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1. What is 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:

\[ r_i = \frac{2 \times l_e(Pyramid)}{\sqrt{5}} \]

Where:

\( r_i \) — Insphere Radius of Tetrakis Hexahedron (m)
\( l_e(Pyramid) \) — Pyramidal Edge Length of Tetrakis Hexahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the pyramidal edge length of the Tetrakis Hexahedron, using the mathematical constant √5.

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 Tetrakis Hexahedron, which has applications in packaging, material science, and architectural design.

4. Using the Calculator

Tips: Enter the pyramidal edge length in meters. The value must be positive and greater than zero. The calculator will compute the insphere radius using the mathematical formula.

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 attached to each face. It has 24 faces, 36 edges, and 14 vertices.

Q2: How is the insphere radius different from the circumsphere radius?
A: The insphere radius is the radius of the largest sphere that fits inside the polyhedron, while the circumsphere radius is the radius of the smallest sphere that contains the polyhedron.

Q3: What are the practical applications of this calculation?
A: This calculation is used in various fields including crystallography, molecular modeling, architectural design, and any application involving geometric optimization of 3D structures.

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

Q5: What is the significance of √5 in this formula?
A: √5 is a mathematical constant that appears in various geometric relationships, particularly those involving pentagonal symmetry and the golden ratio, which relates to the geometry of the Tetrakis Hexahedron.