Insphere Radius of Triakis Octahedron given Volume Calculator

Formula Used:

\[ r_i = \left( \frac{V}{2-\sqrt{2}} \right)^{1/3} \times \sqrt{\frac{5+2\sqrt{2}}{34}} \]

Insphere Radius of Triakis Octahedron (rᵢ):

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1. What is Insphere Radius of Triakis Octahedron?

The Insphere Radius of a Triakis Octahedron is the radius of the sphere that is contained within the Triakis Octahedron such that all the faces are tangent to the sphere. It represents the largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \left( \frac{V}{2-\sqrt{2}} \right)^{1/3} \times \sqrt{\frac{5+2\sqrt{2}}{34}} \]

Where:

\( r_i \) — Insphere Radius of Triakis Octahedron (m)
\( V \) — Volume of Triakis Octahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: The formula calculates the insphere radius based on the volume of the Triakis Octahedron, using mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing properties, structural stability, and spatial relationships within polyhedral structures.

4. Using the Calculator

Tips: Enter the volume of the Triakis Octahedron in cubic meters. The volume must be a positive value greater than zero 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 obtained by adding a triangular pyramid to each face of a regular octahedron.

Q2: How is this formula derived?
A: The formula is derived from the geometric properties and mathematical relationships between the volume and insphere radius of a Triakis Octahedron.

Q3: What are the units of measurement?
A: The volume should be in cubic meters (m³) and the resulting insphere radius will be in meters (m).

Q4: Can this calculator handle very large or small values?
A: Yes, the calculator can handle a wide range of volume values, but extremely large or small values may be limited by PHP's floating-point precision.

Q5: Is this calculation accurate for all Triakis Octahedrons?
A: This calculation is accurate for regular Triakis Octahedrons where all edges and faces maintain the proper geometric relationships.