Total Surface Area of Hexakis Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + (107 \times \sqrt{5}))} \times \left(\frac{\frac{6}{5}}{RA/V}\right)^2 \times \frac{10 \times (417 + (107 \times \sqrt{5}))}{6 \times (185 + (82 \times \sqrt{5}))} \]

Total Surface Area (TSA):

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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 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 formula:

Where:

\( TSA \) — Total Surface Area (m²)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

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

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material science, architectural design, and geometric analysis. It helps in determining coating requirements, heat transfer properties, and structural characteristics of the polyhedron.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. 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: Why is the surface to volume ratio important?
A: The surface to volume ratio is a critical parameter that influences various physical properties including heat transfer, chemical reactivity, and mechanical strength of geometric structures.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size and proportions of the Hexakis Icosahedron, but typically ranges based on the specific geometric configuration.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed exclusively for the Hexakis Icosahedron. Other polyhedra have different geometric relationships and require different formulas.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the Hexakis Icosahedron, assuming exact input values and proper implementation of the formula.