Insphere Radius Of Triakis Icosahedron Given Total Surface Area Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4} \times \sqrt{\frac{11 \times TSA}{15 \times \sqrt{109-30\sqrt{5}}}}} \]

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:

\[ r_i = \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4} \times \sqrt{\frac{11 \times TSA}{15 \times \sqrt{109-30\sqrt{5}}}}} \]

Where:

\( r_i \) — Insphere Radius of Triakis Icosahedron (m)
\( TSA \) — Total Surface Area of Triakis Icosahedron (m²)

Explanation: The formula calculates the insphere radius based on the total surface area of the Triakis Icosahedron, incorporating 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 spatial properties of polyhedra, determining packing efficiency, and analyzing the internal structure of crystalline formations.

4. Using the Calculator

Tips: Enter the total surface area of the Triakis Icosahedron in square meters. The value must be positive and valid. The calculator will compute the insphere radius based on the provided surface area.

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 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 crystallography, nanotechnology, architectural design, and any field that involves geometric modeling of complex polyhedral structures.

Q4: Are there limitations to this formula?
A: The formula is specific to the Triakis Icosahedron geometry and assumes perfect mathematical form. Real-world applications may require adjustments for material properties and manufacturing tolerances.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Triakis Icosahedron. Other polyhedra have different formulas for calculating their insphere radii.