Total Surface Area Of Great Icosahedron Given Circumsphere Radius Calculator

Total Surface Area of Great Icosahedron Formula:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{4 \times r_c}{\sqrt{50 + (22 \times \sqrt{5})}} \right)^2 \]

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 a Great Icosahedron is the total area of all its faces. It is a complex polyhedron with intersecting triangular faces, and its surface area can be calculated using the circumsphere radius.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{4 \times r_c}{\sqrt{50 + (22 \times \sqrt{5})}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Great Icosahedron (m²)
\( r_c \) — Circumsphere Radius of Great Icosahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: The formula derives from the geometric properties of the Great Icosahedron, relating its surface area to the radius of its circumscribed sphere.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of complex polyhedra like the Great Icosahedron is important in geometry, architecture, and materials science for understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the circumsphere radius in 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: 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 this different from a regular icosahedron?
A: While both have 20 triangular faces, the Great Icosahedron is non-convex with self-intersecting faces, whereas a regular icosahedron is convex.

Q3: What are practical applications of this calculation?
A: This calculation is used in mathematical modeling, architectural design of complex structures, and in understanding geometric properties of polyhedra.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Icosahedron. Other polyhedra have different surface area formulas.

Q5: What units should be used for the input?
A: The calculator uses meters for the circumsphere radius, but any consistent unit can be used as long as the surface area units match (e.g., cm² if radius in cm).