Volume of Deltoidal Hexecontahedron given Total Surface Area Calculator

Formula Used:

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

Volume of Deltoidal Hexecontahedron:

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1. What is Volume of Deltoidal Hexecontahedron?

The Volume of Deltoidal Hexecontahedron is the quantity of three dimensional space enclosed by the entire surface of Deltoidal Hexecontahedron. It is a complex polyhedron with 60 deltoid faces and can be calculated from its total surface area using a specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Deltoidal Hexecontahedron (m³)
\( TSA \) — Total Surface Area of Deltoidal Hexecontahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

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

3. Importance of Volume Calculation

Details: Calculating the volume of complex polyhedra like the deltoidal hexecontahedron is important in geometry, crystallography, and materials science for understanding spatial properties and packing efficiency.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and non-zero. The calculator will compute the corresponding volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a deltoidal hexecontahedron?
A: A deltoidal hexecontahedron is a Catalan solid with 60 deltoid (kite-shaped) faces, 120 edges, and 62 vertices.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the deltoidal hexecontahedron, which involves irrational numbers and requires precise mathematical relationships between surface area and volume.

Q3: What are the practical applications of this calculation?
A: This calculation is used in mathematical research, architectural design, and the study of crystal structures and molecular formations.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal hexecontahedron. Other polyhedra have different volume-to-surface-area relationships.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact, though computational precision depends on the implementation and input accuracy.