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Long Edge Of Pentagonal Hexecontahedron Given Total Surface Area Calculator

Formula Used:

\[ Long Edge = \left( \sqrt{\frac{TSA \times (1-2 \times 0.4715756^2)}{30 \times (2+3 \times 0.4715756) \times \sqrt{1-0.4715756^2}}} \times \sqrt{2+2 \times 0.4715756} \right) \times \frac{(7 \times \phi + 2) + (5 \times \phi - 3) + 2 \times (8-3 \times \phi)}{31} \]

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1. What is the Long Edge of Pentagonal Hexecontahedron?

The Long Edge of Pentagonal Hexecontahedron is the length of longest edge which is the top edge of the axial-symmetric pentagonal faces of Pentagonal Hexecontahedron. It's an important geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ Long Edge = \left( \sqrt{\frac{TSA \times (1-2 \times 0.4715756^2)}{30 \times (2+3 \times 0.4715756) \times \sqrt{1-0.4715756^2}}} \times \sqrt{2+2 \times 0.4715756} \right) \times \frac{(7 \times \phi + 2) + (5 \times \phi - 3) + 2 \times (8-3 \times \phi)}{31} \]

Where:

Explanation: This complex formula accounts for the unique geometric properties of the Pentagonal Hexecontahedron, relating its total surface area to the length of its longest edge.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is crucial for understanding the geometry of Pentagonal Hexecontahedron, which has applications in crystallography, molecular modeling, and advanced architectural design.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the corresponding long edge length.

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 golden ratio used in this calculation?
A: The golden ratio appears naturally in the geometry of pentagonal structures and is fundamental to the proportions of the Pentagonal Hexecontahedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in advanced geometry, crystallography, molecular modeling of complex structures, and architectural design of complex forms.

Q4: How accurate is this formula?
A: The formula is mathematically precise for ideal Pentagonal Hexecontahedrons, derived from their geometric properties and relationships.

Q5: Can this calculator handle different units?
A: The calculator uses meters for length and square meters for area. Convert other units to these before calculation for accurate results.

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