Total Surface Area Of Deltoidal Hexecontahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{9}{11} \times \sqrt{10 \times (157 + 31 \times \sqrt{5})} \times \left( \frac{2 \times r_i}{3 \times \sqrt{\frac{135 + 59 \times \sqrt{5}}{205}}} \right)^2 \]

Total Surface Area of Deltoidal Hexecontahedron:

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1. What is Total Surface Area of Deltoidal Hexecontahedron?

The Deltoidal Hexecontahedron is a Catalan solid with 60 deltoid faces. Its total surface area represents the sum of the areas of all its deltoid faces, providing a measure of the complete external coverage of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = \frac{9}{11} \times \sqrt{10 \times (157 + 31 \times \sqrt{5})} \times \left( \frac{2 \times r_i}{3 \times \sqrt{\frac{135 + 59 \times \sqrt{5}}{205}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Deltoidal Hexecontahedron (m²)
\( r_i \) — Insphere Radius of Deltoidal Hexecontahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the total surface area from the insphere 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 analysis, and understanding the geometric properties of this complex polyhedron in mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and non-zero. The calculator will compute the corresponding total surface area of the Deltoidal Hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: It's a Catalan solid with 60 deltoid (kite-shaped) faces, 120 edges, and 62 vertices, known for its complex geometric structure.

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 its faces internally.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the derived formula for the relationship between insphere radius and total surface area.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the base unit. For other units, convert your measurement to meters before input.

Q5: What are practical applications of this calculation?
A: This calculation is useful in crystallography, architectural design, and mathematical research involving polyhedral geometry.