Total Surface Area Of Deltoidal Hexecontahedron Given Midsphere Radius Calculator

Formula Used:

\[ TSA = \frac{9}{11} \times \sqrt{10 \times (157 + (31 \times \sqrt{5})))} \times \left( \frac{20 \times r_m}{3 \times (5 + (3 \times \sqrt{5})))} \right)^2 \]

Total Surface Area of Deltoidal Hexecontahedron:

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1. What is the Total Surface Area of Deltoidal Hexecontahedron?

The Deltoidal Hexecontahedron is a complex polyhedron with 60 deltoid faces. The total surface area represents the sum of the areas of all these deltoid faces, providing a measure of the complete external coverage of the shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{9}{11} \times \sqrt{10 \times (157 + (31 \times \sqrt{5})))} \times \left( \frac{20 \times r_m}{3 \times (5 + (3 \times \sqrt{5})))} \right)^2 \]

Where:

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

Explanation: This formula derives the surface area from the midsphere radius using geometric relationships specific to the deltoidal hexecontahedron's structure.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is essential for various applications including material estimation, heat transfer calculations, and understanding the geometric properties of this complex polyhedron in mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and non-zero. The calculator will compute the total surface area based on the provided midsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: A Deltoidal Hexecontahedron is a polyhedron with 60 deltoid (kite-shaped) faces, 120 edges, and 62 vertices.

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

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the deltoidal hexecontahedron, assuming precise input values.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal hexecontahedron due to its unique geometric properties.

Q5: What are practical applications of this calculation?
A: Applications include architectural design, material science, 3D modeling, and mathematical research involving complex polyhedra.