Insphere Radius of Hexakis Icosahedron given Medium Edge Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \frac{22 \cdot l_{medium}}{3 \cdot (4 + \sqrt{5})} \]

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 following formula:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \frac{22 \cdot l_{medium}}{3 \cdot (4 + \sqrt{5})} \]

Where:

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

3. Formula Explanation

Details: The formula calculates the radius of the largest sphere that can be inscribed within a Hexakis Icosahedron, given the length of its medium edge. It incorporates mathematical constants and geometric relationships specific to this polyhedron.

4. Using the Calculator

Tips: Enter the Medium Edge length 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: What is the significance of the insphere radius?
A: The insphere radius indicates the size of the largest sphere that can fit perfectly inside the polyhedron, touching all its faces internally.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Hexakis Icosahedron, provided the input is accurate.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Icosahedron and its insphere radius calculation.

Q5: What units should be used for the input?
A: The calculator uses meters as the default unit, but the formula works with any consistent unit system (the result will be in the same units as the input).