Insphere Radius of Triakis Octahedron given Pyramidal Edge Length Calculator

Formula Used:

\[ r_i = \frac{l_{e(Pyramid)}}{2-\sqrt{2}} \times \sqrt{\frac{5+2\sqrt{2}}{34}} \]

Insphere Radius of Triakis Octahedron:

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1. What is the Insphere Radius of Triakis Octahedron?

The Insphere Radius of Triakis Octahedron is the radius of the sphere that is contained by the Triakis Octahedron in such a way that all the faces are touching the sphere. It represents the largest sphere that can fit inside the polyhedron while being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{l_{e(Pyramid)}}{2-\sqrt{2}} \times \sqrt{\frac{5+2\sqrt{2}}{34}} \]

Where:

\( r_i \) — Insphere Radius of Triakis Octahedron (m)
\( l_{e(Pyramid)} \) — Pyramidal Edge Length of Triakis Octahedron (m)

Explanation: This formula calculates the radius of the inscribed sphere based on the pyramidal edge length of the Triakis Octahedron, using mathematical constants and square root functions.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing properties, volume relationships, and spatial characteristics of polyhedral structures.

4. Using the Calculator

Tips: Enter the pyramidal edge length in meters. The value must be positive and valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Octahedron?
A: A Triakis Octahedron is a Catalan solid that can be seen as an octahedron with a pyramid added to each face, creating a polyhedron with 24 isosceles triangular faces.

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 and touches all faces, 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 crystallography, molecular modeling, architectural design, and in understanding the geometric properties of various natural and synthetic structures.

Q4: Are there limitations to this formula?
A: This formula is specifically designed for the Triakis Octahedron and assumes a perfect geometric shape. It may not apply to distorted or irregular variations.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, then convert the result back to your desired unit.