Long Edge Of Pentagonal Icositetrahedron Given Total Surface Area Calculator

Formula Used:

\[ l_{Long} = \frac{\sqrt{[Tribonacci_C]+1}}{2} \times \sqrt{\frac{TSA}{3}} \times \left( \frac{(4 \times [Tribonacci_C])-3}{22 \times ((5 \times [Tribonacci_C])-1)} \right)^{\frac{1}{4}} \]

Long Edge of Pentagonal Icositetrahedron:

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1. What is the Long Edge of Pentagonal Icositetrahedron?

The Long Edge of Pentagonal Icositetrahedron is the length of the longest edge which is the top edge of the axial-symmetric pentagonal faces of Pentagonal Icositetrahedron. It is a key geometric parameter in this complex polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ l_{Long} = \frac{\sqrt{[Tribonacci_C]+1}}{2} \times \sqrt{\frac{TSA}{3}} \times \left( \frac{(4 \times [Tribonacci_C])-3}{22 \times ((5 \times [Tribonacci_C])-1)} \right)^{\frac{1}{4}} \]

Where:

\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)
\( TSA \) — Total Surface Area of Pentagonal Icositetrahedron (m²)

Explanation: This formula relates the long edge length to the total surface area through the Tribonacci constant, which is fundamental to the geometry of pentagonal icositetrahedra.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is essential for understanding the geometric properties, structural integrity, and spatial characteristics of pentagonal icositetrahedra in mathematical and engineering applications.

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 long edge length based on the mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A pentagonal icositetrahedron is a Catalan solid with 24 pentagonal faces, 60 edges, and 38 vertices. It is the dual of the snub cube.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the real root of the equation x³ - x² - x - 1 = 0, approximately equal to 1.839286755214161.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of pentagonal icositetrahedra and the Tribonacci constant relationship.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to pentagonal icositetrahedra due to their unique geometric properties and relationship with the Tribonacci constant.

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, mathematical modeling, and the study of complex geometric structures.