Long Edge of Pentagonal Hexecontahedron Given Midsphere Radius Calculator

Formula Used:

\[ le(Long) = \frac{r_m}{\sqrt{\frac{1+0.4715756}{2(1-2\times0.4715756)}} \times \sqrt{2+2\times0.4715756}} \times \frac{(7\phi+2)+(5\phi-3)+2(8-3\phi)}{31} \]

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 a key geometric parameter in this complex polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ le(Long) = \frac{r_m}{\sqrt{\frac{1+0.4715756}{2(1-2\times0.4715756)}} \times \sqrt{2+2\times0.4715756}} \times \frac{(7\phi+2)+(5\phi-3)+2(8-3\phi)}{31} \]

Where:

\( le(Long) \) — Long Edge of Pentagonal Hexecontahedron (m)
\( r_m \) — Midsphere Radius of Pentagonal Hexecontahedron (m)
\( \phi \) — Golden ratio (≈1.618034)
0.4715756 — Constant value specific to the geometry

3. Formula Explanation

Details: This formula relates the long edge of the pentagonal hexecontahedron to its midsphere radius through a complex mathematical relationship involving the golden ratio and specific geometric constants.

4. Using the Calculator

Tips: Enter the midsphere radius in 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 Catalan solid with 60 pentagonal faces, 150 edges, and 92 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: Why is the golden ratio used in the formula?
A: The golden ratio appears naturally in many geometric constructions and is fundamental to the proportions of the pentagonal hexecontahedron.

Q4: What are typical values for the long edge?
A: The long edge length depends on the specific dimensions of the polyhedron and can vary based on the midsphere radius.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the pentagonal hexecontahedron geometry.