Volume Of Hexakis Icosahedron Given Truncated Icosidodecahedron Edge Calculator

Volume of Hexakis Icosahedron Formula:

\[ V = \frac{25}{88} \times \sqrt{6 \times (185 + 82 \times \sqrt{5})} \times \frac{8}{125} \times l_e^3 \times (\sqrt{15 \times (5 - \sqrt{5})})^3 \]

Volume of Hexakis Icosahedron:

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1. What is the Volume of Hexakis Icosahedron?

The volume of a Hexakis Icosahedron is the quantity of three-dimensional space enclosed by the entire surface of this complex polyhedron. It is calculated based on the truncated edge length derived from an Icosidodecahedron.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ V = \frac{25}{88} \times \sqrt{6 \times (185 + 82 \times \sqrt{5})} \times \frac{8}{125} \times l_e^3 \times (\sqrt{15 \times (5 - \sqrt{5})})^3 \]

Where:

\( V \) — Volume of Hexakis Icosahedron (m³)
\( l_e \) — Truncated Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This complex formula accounts for the geometric properties of the Hexakis Icosahedron and its relationship to the truncated Icosidodecahedron edge length.

3. Importance of Volume Calculation

Details: Calculating the volume of complex polyhedra like the Hexakis Icosahedron is important in various fields including crystallography, molecular modeling, architectural design, and mathematical research on geometric properties.

4. Using the Calculator

Tips: Enter the truncated edge length of the Hexakis Icosahedron in meters. The value must be positive and valid for accurate volume 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 Icosidodecahedron, featuring 120 faces, 180 edges, and 62 vertices.

Q2: How is this different from a regular Icosahedron?
A: While both are polyhedra, the Hexakis Icosahedron is much more complex with 120 triangular faces compared to the regular Icosahedron's 20 faces.

Q3: What practical applications does this calculation have?
A: This calculation is used in crystallography, molecular structure analysis, architectural design of complex structures, and mathematical research.

Q4: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Hexakis Icosahedron, which involves golden ratio relationships and multiple geometric transformations.

Q5: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of positive values, but extremely large values may cause computational limitations in the browser environment.