Medium Edge of Hexakis Icosahedron given Total Surface Area Calculator

Formula Used:

\[ l_{Medium} = \frac{3}{22} \times (4+\sqrt{5}) \times \sqrt{\frac{44 \times TSA}{15 \times \sqrt{10 \times (417+107\sqrt{5})}}} \]

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 one of the three distinct edge lengths in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Medium} = \frac{3}{22} \times (4+\sqrt{5}) \times \sqrt{\frac{44 \times TSA}{15 \times \sqrt{10 \times (417+107\sqrt{5})}}} \]

Where:

\( l_{Medium} \) — Medium Edge 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 medium edge length based on the total surface area of the Hexakis Icosahedron, using mathematical constants derived from its geometric properties.

3. Importance of Medium Edge Calculation

Details: Calculating the medium edge is important for understanding the geometry and proportions of the Hexakis Icosahedron, which has applications in crystallography, molecular modeling, and architectural design.

4. Using the Calculator

Tips: Enter the total surface area 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: How many different edge lengths does a Hexakis Icosahedron have?
A: A Hexakis Icosahedron has three distinct edge lengths: short, medium, and long edges.

Q3: What are typical applications of Hexakis Icosahedron calculations?
A: These calculations are used in geometric modeling, crystallography, architectural design, and in the study of complex polyhedral structures.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula. The accuracy of the result depends on the precision of the input value.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Hexakis Icosahedron only. Other polyhedra have different geometric relationships and require different formulas.