Medium Edge Of Hexakis Icosahedron Given Insphere Radius Calculator

Formula Used:

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

Medium Edge of Hexakis Icosahedron:

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1. What is the Medium Edge of Hexakis Icosahedron?

The Medium Edge of Hexakis Icosahedron is the length of the edge that connects two non-adjacent and non-opposite vertices of the Hexakis Icosahedron. It is an important geometric parameter in this complex polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the medium edge length based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the Hexakis Icosahedron.

3. Importance of Medium Edge Calculation

Details: Calculating the medium edge is crucial for understanding the geometric properties of Hexakis Icosahedron, including surface area, volume calculations, and spatial relationships within this complex polyhedral structure.

4. Using the Calculator

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

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 insphere radius?
A: The insphere radius is the radius of the largest sphere that can be contained within the Hexakis Icosahedron, touching all its faces.

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

Q4: 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, then convert the result back to your desired unit.

Q5: What are typical values for the insphere radius?
A: The insphere radius depends on the specific dimensions of the Hexakis Icosahedron. For standard configurations, it typically ranges from fractions of a meter to several meters.