Insphere Radius of Hexakis Icosahedron Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}}{4} \times l_e \]

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:

\[ r_i = \frac{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}}{4} \times l_e \]

Where:

\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( l_e \) — Long Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the insphere radius based on the long edge length of the Hexakis 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 3D modeling for understanding the spatial properties of the Hexakis Icosahedron, determining packing efficiency, and analyzing the relationship between different geometric parameters of the shape.

4. Using the Calculator

Tips: Enter the Long Edge of Hexakis Icosahedron 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 typical values for the insphere radius?
A: The insphere radius depends on the size of the Hexakis Icosahedron. For a unit long edge, the insphere radius is approximately 0.14508 meters.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Hexakis Icosahedron only. Other polyhedra have different formulas for calculating their insphere radii.

Q5: What practical applications does this calculation have?
A: This calculation is used in mathematical geometry, 3D modeling, crystallography, and architectural design where Hexakis Icosahedron shapes are employed.