Total Surface Area of Hexakis Icosahedron given Truncated Icosidodecahedron Edge Calculator

Formula Used:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \frac{4}{25} \times l_{e(Truncated\ Icosidodecahedron)}^2 \times 15 \times (5 - \sqrt{5}) \]

Total Surface Area of Hexakis Icosahedron:

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1. What is the Total Surface Area of Hexakis Icosahedron?

The Total Surface Area of a Hexakis Icosahedron is the total area of all its faces. It is a measure of the two-dimensional space covered by the surface of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \frac{4}{25} \times l_{e}^2 \times 15 \times (5 - \sqrt{5}) \]

Where:

\( TSA \) — Total Surface Area of Hexakis Icosahedron (m²)
\( l_{e} \) — Truncated Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

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

3. Importance of Surface Area Calculation

Details: Calculating the surface area is important in various fields including geometry, architecture, material science, and 3D modeling where understanding the surface properties of complex shapes is required.

4. Using the Calculator

Tips: Enter the truncated edge length in meters. The value must be positive and valid 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. It has 120 faces, 180 edges, and 62 vertices.

Q2: How is this different from a regular Icosahedron?
A: While both are polyhedra, a regular icosahedron has 20 equilateral triangle faces, whereas a Hexakis Icosahedron has 120 scalene triangle faces.

Q3: What are practical applications of this calculation?
A: This calculation is useful in crystallography, architectural design, and the study of complex geometric structures.

Q4: How accurate is this formula?
A: The formula is mathematically precise for ideal geometric shapes and provides exact surface area calculations.

Q5: Can this calculator handle very large or small values?
A: Yes, the calculator can process a wide range of values, though extremely large values may be limited by computational precision.