Insphere Radius of Hexakis Icosahedron given Short Edge Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \frac{44 \cdot l_{short}}{5 \cdot (7 - \sqrt{5})} \]

Insphere Radius of Hexakis Icosahedron:

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1. What is Insphere Radius of Hexakis Icosahedron?

The Insphere Radius of 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 maximum sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \frac{44 \cdot l_{short}}{5 \cdot (7 - \sqrt{5})} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( l_{short} \) — Short Edge of Hexakis Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the radius of the inscribed sphere based on the length of the shortest edge of the Hexakis Icosahedron, using mathematical constants and geometric relationships.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and 3D modeling for understanding the spatial properties of polyhedra, determining packing efficiency, and analyzing geometric relationships within complex shapes.

4. Using the Calculator

Tips: Enter the length of the short edge in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.

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: 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, molecular modeling, architectural design, and computer graphics where precise geometric properties of complex shapes are required.

Q4: How accurate is this formula?
A: The formula is mathematically exact for a perfect Hexakis Icosahedron and provides precise results when correct input values are used.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, or convert the result from meters to your desired unit.