Truncated Icosidodecahedron Edge of Hexakis Icosahedron given Total Surface Area Calculator

Formula Used:

\[ le = \frac{5}{2 \times \sqrt{15 \times (5 - \sqrt{5})}} \times \sqrt{\frac{44 \times TSA}{15 \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}} \]

Truncated Edge Length (le):

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1. What is the Truncated Icosidodecahedron Edge of Hexakis Icosahedron?

The Truncated Edge of Hexakis Icosahedron is the length of the edges of a Hexakis Icosahedron that is created by truncating the vertices of an Icosidodecahedron. This geometric measurement is important in understanding the properties of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ le = \frac{5}{2 \times \sqrt{15 \times (5 - \sqrt{5})}} \times \sqrt{\frac{44 \times TSA}{15 \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}} \]

Where:

\( le \) — Truncated Edge Length (meters)
\( TSA \) — Total Surface Area (square meters)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the truncated edge length based on the total surface area of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of This Calculation

Details: Calculating the truncated edge length is crucial for geometric analysis, architectural design applications, and understanding the spatial properties of complex polyhedra in mathematical and engineering contexts.

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 icosahedron, featuring 120 faces, 180 edges, and 62 vertices.

Q2: What units should I use for the input?
A: The calculator expects total surface area in square meters, and returns the edge length in meters.

Q3: Can this calculator handle very large or very small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may affect precision due to floating-point arithmetic limitations.

Q4: What is the geometric significance of the constants in the formula?
A: The constants (5, 15, 44, 417, 107) are derived from the geometric properties and trigonometric relationships specific to the Hexakis Icosahedron structure.

Q5: Are there any limitations to this calculation?
A: This calculation assumes a perfect geometric form and may not account for manufacturing tolerances or material properties in practical applications.