Long Edge Of Pentagonal Hexecontahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ Long Edge = \frac{6(2+3k)\sqrt{1-k^2}}{1-2k^2} \div \left( SA:V \times \frac{(1+k)(2+3k)}{(1-2k^2)\sqrt{1-2k}} \right) \times \sqrt{2+2k} \times \frac{(7\phi+2)+(5\phi-3)+2(8-3\phi)}{31} \]

where \( k = 0.4715756 \) and \( \phi = 1.61803398874989484820458683436563811 \) (Golden Ratio)

Long Edge of Pentagonal Hexecontahedron:

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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 a key geometric parameter of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ Long Edge = \frac{6(2+3k)\sqrt{1-k^2}}{1-2k^2} \div \left( SA:V \times \frac{(1+k)(2+3k)}{(1-2k^2)\sqrt{1-2k}} \right) \times \sqrt{2+2k} \times \frac{(7\phi+2)+(5\phi-3)+2(8-3\phi)}{31} \]

Where:

\( k = 0.4715756 \) — A constant parameter
\( \phi = 1.61803398874989484820458683436563811 \) — Golden Ratio
\( SA:V \) — Surface to Volume Ratio (input value)
All calculations use precise mathematical operations

Explanation: This complex formula relates the long edge length to the surface-to-volume ratio through geometric relationships specific to the pentagonal hexecontahedron.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is essential for understanding the geometry of pentagonal hexecontahedrons, which have applications in crystallography, material science, and advanced geometry studies.

4. Using the Calculator

Tips: Enter the surface-to-volume ratio in m⁻¹. The value must be positive and valid for meaningful results.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is the golden ratio used in this formula?
A: The golden ratio appears naturally in the geometry of pentagonal structures and provides the most efficient and aesthetically pleasing proportions.

Q3: What are typical values for surface-to-volume ratio?
A: This depends on the specific dimensions of the polyhedron. Smaller polyhedrons have higher SA:V ratios, while larger ones have lower ratios.

Q4: Can this calculator be used for other polyhedrons?
A: No, this specific formula is designed only for the pentagonal hexecontahedron due to its unique geometric properties.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise when using the exact constants provided. Rounding may occur in the final displayed result.