Long Ridge Length of Great Icosahedron given Total Surface Area Calculator

Formula Used:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \sqrt{\frac{TSA}{3\sqrt{3} \cdot (5 + 4\sqrt{5})}} \]

Long Ridge Length of Great Icosahedron:

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1. What is the Long Ridge Length of Great Icosahedron?

The Long Ridge Length of Great Icosahedron is the length of any of the edges that connects the peak vertex and adjacent vertex of the pentagon on which each peak of Great Icosahedron is attached. It is an important geometric measurement in this complex polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \sqrt{\frac{TSA}{3\sqrt{3} \cdot (5 + 4\sqrt{5})}} \]

Where:

\( l_{Ridge(Long)} \) — Long Ridge Length of Great Icosahedron (meters)
\( TSA \) — Total Surface Area of Great Icosahedron (square meters)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the long ridge length based on the total surface area of the great icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Long Ridge Length Calculation

Details: Calculating the long ridge length is essential for understanding the geometric properties of the great icosahedron, which has applications in mathematical modeling, architectural design, and complex geometric analysis.

4. Using the Calculator

Tips: Enter the total surface area of the great icosahedron in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The great icosahedron is one of the four regular star polyhedra, consisting of 20 triangular faces that intersect each other in a complex pattern.

Q2: How is the long ridge length different from other edges?
A: The long ridge specifically refers to the edges connecting peak vertices to adjacent vertices of the pentagonal structures within the polyhedron.

Q3: What units should I use for input?
A: The calculator expects the total surface area in square meters, and returns the long ridge length in meters.

Q4: Are there limitations to this calculation?
A: The formula assumes a perfect great icosahedron geometry and may not apply to distorted or irregular variations of the shape.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed exclusively for calculating the long ridge length of the great icosahedron based on its total surface area.