Insphere Radius of Triakis Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4} \times \frac{12\sqrt{109-30\sqrt{5}}}{(5+7\sqrt{5}) \times \frac{S}{V}} \]

Insphere Radius (rᵢ):

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1. What is Insphere Radius of Triakis Icosahedron?

The Insphere Radius of a Triakis Icosahedron is the radius of the largest sphere that can be contained within the polyhedron such that it touches all faces tangentially. It represents the distance from the center to the closest point on any face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4} \times \frac{12\sqrt{109-30\sqrt{5}}}{(5+7\sqrt{5}) \times \frac{S}{V}} \]

Where:

\( r_i \) — Insphere radius (m)
\( \frac{S}{V} \) — Surface to volume ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the insphere radius based on the surface to volume ratio of the Triakis Icosahedron, incorporating mathematical constants related to its geometric properties.

3. Importance of Insphere Radius Calculation

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

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and valid for accurate calculation of the insphere radius.

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, featuring 60 isosceles triangular faces.

Q2: How is surface to volume ratio defined?
A: Surface to volume ratio is the total surface area divided by the total volume of the polyhedron, measured in 1/m.

Q3: What are typical values for insphere radius?
A: The insphere radius varies depending on the size and proportions of the Triakis Icosahedron, typically ranging from fractions of a meter to several meters.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect Triakis Icosahedron shape and may not account for manufacturing tolerances or material properties.

Q5: What applications use this calculation?
A: This calculation is used in crystallography, nanotechnology, architectural design, and any field dealing with polyhedral structures and their geometric properties.