Edge Length of Great Icosahedron given Total Surface Area Calculator

Formula Used:

\[ Edge\ Length\ of\ Great\ Icosahedron = \sqrt{\frac{Total\ Surface\ Area\ of\ Great\ Icosahedron}{3 \times \sqrt{3} \times (5 + (4 \times \sqrt{5}))}} \]

Edge Length of Great Icosahedron:

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1. What is Edge Length of Great Icosahedron?

The Edge Length of Great Icosahedron is the distance between any pair of adjacent peak vertices of the Great Icosahedron. It is a fundamental geometric measurement used to define the size and proportions of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ Edge\ Length = \sqrt{\frac{Total\ Surface\ Area}{3 \times \sqrt{3} \times (5 + (4 \times \sqrt{5}))}} \]

Where:

Total Surface Area — The total surface area of the Great Icosahedron (m²)
√3 — Square root of 3 (approximately 1.732)
√5 — Square root of 5 (approximately 2.236)

Explanation: This formula derives from the geometric properties of the Great Icosahedron, relating the edge length to the total surface area through mathematical constants and relationships specific to this polyhedron.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the scale and proportions of the Great Icosahedron. It helps in geometric modeling, architectural design, and mathematical analysis of this complex three-dimensional shape.

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. The calculator will compute the corresponding edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is one of the four Kepler-Poinsot polyhedra, consisting of 20 triangular faces that intersect each other, creating a complex star-shaped polyhedron.

Q2: How is this different from a regular icosahedron?
A: Unlike the regular icosahedron where faces don't intersect, the Great Icosahedron has self-intersecting faces, creating a more complex structure with star-shaped properties.

Q3: What are typical edge length values?
A: Edge length values depend on the scale of the polyhedron. They can range from microscopic scales to architectural sizes, depending on the application.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Icosahedron due to its unique geometric properties and surface area relationships.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is suitable for most mathematical and engineering applications involving the Great Icosahedron.