Insphere Radius of Triakis Icosahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4} \times \left( \frac{44V}{5(5+7\sqrt{5})} \right)^{\frac{1}{3}} \]

Insphere Radius of Triakis Icosahedron:

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1. What is the Insphere Radius of Triakis Icosahedron?

The Insphere Radius of Triakis Icosahedron is the radius of the sphere that is contained by the Triakis Icosahedron 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:

\[ r_i = \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4} \times \left( \frac{44V}{5(5+7\sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

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

Explanation: This formula calculates the insphere radius based on the volume of the Triakis Icosahedron, using mathematical constants derived from the geometric properties of this polyhedron.

3. Importance of Insphere Radius Calculation

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

4. Using the Calculator

Tips: Enter the volume of the Triakis Icosahedron in cubic meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 60 faces, 90 edges, and 32 vertices.

Q2: How is this formula derived?
A: The formula is derived from the geometric relationships between the volume and insphere radius of the Triakis Icosahedron, using mathematical constants specific to this polyhedron's structure.

Q3: What units should I use?
A: Use consistent units (typically meters for length and cubic meters for volume). The calculator will output the radius in the same length unit as the volume's cube root.

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 Icosahedrons?
A: Yes, this formula provides the exact insphere radius for any regular Triakis Icosahedron given its volume.