Total Surface Area Of Great Icosahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5})}{\frac{1}{4} \times (25 + 9 \times \sqrt{5}) \times RA/V} \right)^2 \]

Total Surface Area of Great Icosahedron (TSA):

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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 all triangular faces that make up the Great Icosahedron's surface.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5})}{\frac{1}{4} \times (25 + 9 \times \sqrt{5}) \times RA/V} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Great Icosahedron (m²)
\( RA/V \) — Surface to Volume Ratio of Great Icosahedron (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the known surface to volume ratio of the Great Icosahedron, utilizing 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 surface to volume ratio value in 1/m. 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, composed of 20 equilateral triangular faces that intersect each other.

Q2: How is this different from a regular icosahedron?
A: While both have 20 triangular faces, the Great Icosahedron is a star polyhedron where the faces intersect, creating a more complex structure than the convex regular icosahedron.

Q3: What are typical surface to volume ratio values for Great Icosahedron?
A: The surface to volume ratio depends on the scale of the polyhedron, but typically ranges from very small values for large objects to larger values for smaller objects.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed exclusively for the Great Icosahedron due to its unique geometric properties.

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