Midsphere Radius Of Hexakis Icosahedron Given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{1}{8} \times \sqrt{\frac{44}{15} \times \frac{TSA}{\sqrt{10 \times (417 + 107 \times \sqrt{5})}}} \times (5 + 3 \times \sqrt{5}) \]

Midsphere Radius (rₘ):

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1. What is Midsphere Radius of Hexakis Icosahedron?

The Midsphere Radius of a Hexakis Icosahedron is defined as the radius of the sphere for which all the edges of the Hexakis Icosahedron become a tangent line on that sphere. It's an important geometric property of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_m = \frac{1}{8} \times \sqrt{\frac{44}{15} \times \frac{TSA}{\sqrt{10 \times (417 + 107 \times \sqrt{5})}}} \times (5 + 3 \times \sqrt{5}) \]

Where:

\( r_m \) — Midsphere Radius (m)
\( TSA \) — Total Surface Area (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives the midsphere radius from the total surface area of a Hexakis Icosahedron, incorporating the mathematical constant √5 which is fundamental to icosahedral geometry.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is crucial for understanding the spatial properties and geometric relationships within a Hexakis Icosahedron. It helps in various applications including 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. The calculator will compute the corresponding midsphere 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: Why is √5 involved in the formula?
A: The square root of 5 appears frequently in formulas related to icosahedral symmetry and the golden ratio, which are fundamental to this polyhedron's geometry.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the size of the polyhedron. For a Hexakis Icosahedron with unit edge length, the midsphere radius is approximately 3.5 units.

Q4: Can this calculator handle very large or small values?
A: Yes, the calculator can process a wide range of surface area values, though extremely large values may be limited by computational precision.

Q5: Is this calculation accurate for all Hexakis Icosahedrons?
A: Yes, the formula is mathematically exact and applies to all regular Hexakis Icosahedrons regardless of size.