Total Surface Area Of Deltoidal Hexecontahedron Given Volume Calculator

Formula Used:

\[ TSA = \frac{9}{11} \times \sqrt{10 \times (157 + 31 \times \sqrt{5})} \times \left( \frac{11 \times V}{45 \times \sqrt{\frac{370 + 164 \times \sqrt{5}}{25}}} \right)^{\frac{2}{3}} \]

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. Its total surface area represents the sum of the areas of all its deltoid faces, providing a measure of the complete exterior surface coverage of this three-dimensional shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = \frac{9}{11} \times \sqrt{10 \times (157 + 31 \times \sqrt{5})} \times \left( \frac{11 \times V}{45 \times \sqrt{\frac{370 + 164 \times \sqrt{5}}{25}}} \right)^{\frac{2}{3}} \]

Where:

\( TSA \) — Total Surface Area of Deltoidal Hexecontahedron (m²)
\( V \) — Volume of Deltoidal Hexecontahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the surface area from the volume using the geometric properties and mathematical relationships specific to the deltoidal hexecontahedron shape.

3. Importance of Total Surface Area Calculation

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

4. Using the Calculator

Tips: Enter the volume of the deltoidal hexecontahedron in cubic meters. The volume must be a positive value greater than zero. The calculator will compute the corresponding total surface area based on the mathematical relationship between volume and surface area for this specific shape.

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. It's one of the Catalan solids, which are the duals of the Archimedean solids.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometric relationships in this 60-faced polyhedron. The formula incorporates mathematical constants and relationships specific to this shape's geometry.

Q3: What units should I use?
A: Use consistent units throughout. If volume is in cubic meters, the surface area result will be in square meters. You can use any unit system as long as you maintain consistency.

Q4: Can this calculator handle very large or very small volumes?
A: Yes, the calculator can handle a wide range of volume values, though extremely large or small values may be limited by computational precision.

Q5: What are practical applications of this calculation?
A: This calculation is useful in crystallography, architectural design, mathematical modeling, and any field dealing with complex polyhedral structures where surface area to volume ratios are important.