Long Edge of Pentagonal Hexecontahedron Given Insphere Radius Calculator

Formula Used:

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

Long Edge of Pentagonal Hexecontahedron:

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1. What is Long Edge of Pentagonal Hexecontahedron?

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

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

\( r_i \) — Insphere Radius of Pentagonal Hexecontahedron (m)
\( \phi \) — Golden ratio (approximately 1.618034)
0.4715756 — Constant coefficient

Explanation: This formula calculates the longest edge length based on the insphere radius and incorporates the golden ratio constant for geometric precision.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is crucial for understanding the geometric properties, symmetry, and spatial dimensions of the Pentagonal Hexecontahedron, which has applications in crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding long edge length of the Pentagonal Hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A Pentagonal Hexecontahedron is a complex polyhedron with 60 pentagonal faces, making it one of the Catalan solids.

Q2: What is the significance of the golden ratio in this formula?
A: The golden ratio (φ) appears naturally in many geometric constructions and provides optimal proportions in the Pentagonal Hexecontahedron's structure.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the given formula, with results rounded to 6 decimal places for practical use.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Pentagonal Hexecontahedron and its long edge calculation.

Q5: What units should I use for the input?
A: The input should be in meters, and the result will be in meters. You can convert from other units as needed before calculation.