Total Surface Area of Great Icosahedron given Long Ridge Length Calculator

Formula Used:

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

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 Great Icosahedron refers to the total area covered by all the faces of the Great Icosahedron. It is a geometric measurement that helps in understanding the extent of the surface of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area of Great Icosahedron (m²)
\( l_{Ridge(Long)} \) — Long Ridge Length of Great Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the total surface area based on the long ridge length, incorporating geometric constants and square roots to account for the complex structure of the Great Icosahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is crucial for various applications in geometry, architecture, and material science, where understanding the extent of a surface is necessary for design, analysis, and resource estimation.

4. Using the Calculator

Tips: Enter the long ridge length in meters. The value must be positive and greater than zero to compute the total surface area accurately.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is a non-convex polyhedron with 20 triangular faces. It is one of the Kepler-Poinsot solids and has a complex star-like structure.

Q2: Why is the surface area important?
A: Surface area calculations are essential in fields like material science for determining coating requirements, in architecture for design purposes, and in mathematics for understanding geometric properties.

Q3: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Great Icosahedron using its unique geometric properties and formulas.

Q4: What units are used in the calculation?
A: The calculator uses meters for length and square meters for area. Ensure consistent units for accurate results.

Q5: Are there limitations to this calculation?
A: The calculation assumes ideal geometric conditions and may not account for real-world variations or imperfections in the shape.