Total Surface Area Of Pentagonal Hexecontahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ TSA = 30 \times \left( \frac{6 \times (2+3 \times 0.4715756) \times \sqrt{1-0.4715756^2}/(1-2 \times 0.4715756^2)}{AV \times \frac{(1+0.4715756) \times (2+3 \times 0.4715756)}{(1-2 \times 0.4715756^2) \times \sqrt{1-2 \times 0.4715756}}} \right)^2 \times \frac{(2+3 \times 0.4715756) \times \sqrt{1-0.4715756^2}}{1-2 \times 0.4715756^2} \]

Total Surface Area of Pentagonal Hexecontahedron:

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1. What is Total Surface Area of Pentagonal Hexecontahedron?

The Total Surface Area of a Pentagonal Hexecontahedron is the total area of all faces of this complex polyhedron. It's an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

Where:

\( TSA \) — Total Surface Area of Pentagonal Hexecontahedron (m²)
\( AV \) — Surface to Volume Ratio of Pentagonal Hexecontahedron (1/m)
0.4715756 — Constant value used in the calculation

Explanation: This complex formula calculates the total surface area based on the surface to volume ratio, using specific geometric properties of the pentagonal hexecontahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area of complex polyhedra is crucial in various fields including materials science, architecture, and 3D modeling where surface properties affect material behavior and design considerations.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A pentagonal hexecontahedron is a complex polyhedron with 60 pentagonal faces. It's a Catalan solid, the dual of the snub dodecahedron.

Q2: Why is the constant 0.4715756 used in the formula?
A: This constant represents specific geometric relationships and ratios inherent to the pentagonal hexecontahedron's structure.

Q3: What are typical surface area values for this shape?
A: The surface area varies significantly based on the size of the polyhedron. There's no "normal" value as it depends on the specific dimensions.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the pentagonal hexecontahedron. Other polyhedra require different formulas.

Q5: What units should I use for the surface to volume ratio?
A: The calculator expects the surface to volume ratio in 1/m (inverse meters), which is the standard SI unit for this measurement.