Total Surface Area Of Hexakis Icosahedron Given Midsphere Radius Calculator

Formula Used:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \left( \frac{8 \times r_m}{5 + 3 \times \sqrt{5}} \right)^2 \]

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 polyhedron with 120 triangular faces, 180 edges, and 62 vertices. The surface area calculation is important in geometry and various practical applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \left( \frac{8 \times r_m}{5 + 3 \times \sqrt{5}} \right)^2 \]

Where:

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

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

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric shapes is fundamental in mathematics, engineering, architecture, and various scientific fields. It helps in material estimation, structural analysis, and understanding geometric properties.

4. Using the Calculator

Tips: Enter the midsphere radius in 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 dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of a sphere that is tangent to all edges of the polyhedron.

Q3: What are typical values for surface area?
A: The surface area depends on the size of the polyhedron. For a given midsphere radius, the surface area can be calculated using the provided formula.

Q4: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Hexakis Icosahedron. Other polyhedra require different formulas.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact based on the formula. The precision depends on the input value and the implementation of mathematical functions.