Total Surface Area of Great Icosahedron given Short Ridge Length Calculator

Total Surface Area of Great Icosahedron Formula:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{5 \times l_{Ridge(Short)}}{\sqrt{10}} \right)^2 \]

Total Surface Area of Great Icosahedron:

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1. What is the Total Surface Area of Great Icosahedron?

The Total Surface Area of a Great Icosahedron is the total quantity of plane enclosed on the entire surface of this complex polyhedron. It represents the sum of the areas of all its triangular faces.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{5 \times l_{Ridge(Short)}}{\sqrt{10}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Great Icosahedron (m²)
\( l_{Ridge(Short)} \) — Short Ridge Length of Great Icosahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)
\( \sqrt{10} \) — Square root of 10 (approximately 3.162)

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

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for understanding the geometric properties of the great icosahedron, material requirements for physical models, and various applications in mathematics, architecture, and engineering where this complex shape is utilized.

4. Using the Calculator

Tips: Enter the short ridge length measurement in meters. The value must be positive and greater than zero. The calculator will compute the total surface area using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: A Great Icosahedron is a non-convex polyhedron with 20 triangular faces that intersect each other, creating a complex star-shaped structure.

Q2: How is the short ridge length defined?
A: The short ridge length is defined as the maximum vertical distance between the finished bottom level and the finished top height directly above of the Great Icosahedron.

Q3: What units should be used for input?
A: The calculator expects input in meters, and the result will be in square meters. For other units, convert appropriately before calculation.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric form of the great icosahedron. For physical objects, manufacturing tolerances may affect the actual surface area.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is designed only for the great icosahedron. Other polyhedra have different surface area formulas based on their unique geometric properties.