Total Surface Area of Hexakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \frac{(4 \times r_i)^2}{\frac{15}{241} \times (275 + 119 \times \sqrt{5})} \]

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's a geometric measurement that represents the amount of two-dimensional space covered by the surface of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \frac{(4 \times r_i)^2}{\frac{15}{241} \times (275 + 119 \times \sqrt{5})} \]

Where:

\( TSA \) — Total Surface Area of Hexakis Icosahedron (m²)
\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

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

3. Importance of Surface Area Calculation

Details: Calculating the surface area of complex polyhedra is crucial in various fields including materials science, architecture, and 3D modeling. It helps in determining material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the total surface area based on the mathematical formula.

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: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, touching all faces tangentially.

Q3: What units should I use for the input?
A: The calculator uses meters for the insphere radius and returns square meters for the surface area. Ensure consistent units throughout your calculations.

Q4: How accurate is this calculation?
A: The calculation is mathematically precise based on the given formula. The result is rounded to 6 decimal places for practical use.

Q5: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.