Total Surface Area of Truncated Icosahedron Calculator

Total Surface Area Formula:

\[ TSA = 3 \times le^2 \times ((10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})}) \]

Total Surface Area (TSA):

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

The Total Surface Area of a Truncated Icosahedron is the total quantity of plane enclosed by the entire surface of this polyhedron. A truncated icosahedron is an Archimedean solid with 32 faces - 12 regular pentagons and 20 regular hexagons.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = 3 \times le^2 \times ((10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})}) \]

Where:

\( TSA \) — Total Surface Area (m²)
\( le \) — Edge Length of the truncated icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area by summing the areas of all 32 faces (12 pentagons and 20 hexagons) that make up the truncated icosahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is fundamental in various fields including architecture, materials science, chemistry (for surface reactions), and engineering. For truncated icosahedrons specifically, this is particularly relevant in studying fullerene molecules and architectural designs.

4. Using the Calculator

Tips: Enter the edge length of the truncated icosahedron in meters. The value must be positive and greater than zero. The calculator will compute the total surface area in square meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid obtained by truncating the vertices of a regular icosahedron. It has 32 faces, 90 edges, and 60 vertices.

Q2: Where is this shape commonly found?
A: The truncated icosahedron is best known as the shape of a soccer ball. It's also the structure of Buckminsterfullerene (C60) molecules in chemistry.

Q3: What are the units for the result?
A: The result is in square meters (m²), which is the standard SI unit for area. If you input edge length in other units, the result will be in those units squared.

Q4: Can this calculator handle very small or very large values?
A: Yes, the calculator can handle a wide range of values, though extremely large values may be limited by PHP's floating-point precision.

Q5: Is this formula exact or approximate?
A: This is an exact mathematical formula for calculating the total surface area of a perfect truncated icosahedron.