Insphere Radius of Hexakis Icosahedron given Total Surface Area Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \sqrt{\frac{44 \cdot TSA}{15 \cdot \sqrt{10 \cdot (417 + 107\sqrt{5})}}} \]

Insphere Radius of Hexakis Icosahedron:

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1. What is the Insphere Radius of Hexakis Icosahedron?

The Insphere Radius of a Hexakis Icosahedron is defined as the radius of the sphere that is contained by the Hexakis 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:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \sqrt{\frac{44 \cdot TSA}{15 \cdot \sqrt{10 \cdot (417 + 107\sqrt{5})}}} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( TSA \) — Total Surface Area of Hexakis Icosahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the total surface area of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Insphere Radius Calculation

Details: The insphere radius is important in geometry and material science for understanding the internal packing properties, volume optimization, and spatial relationships within complex polyhedral structures.

4. Using the Calculator

Tips: Enter the total surface area of the Hexakis Icosahedron in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Hexakis Icosahedron, which requires precise mathematical relationships between its various dimensions and properties.

Q3: What are typical values for the insphere radius?
A: The insphere radius depends on the size of the polyhedron. For a given total surface area, the radius can be calculated using this specific formula.

Q4: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Hexakis Icosahedron. Other polyhedra have different formulas for calculating their insphere radii.

Q5: What units should I use?
A: The calculator uses meters for length and square meters for area. Ensure consistent units for accurate results.