Total Surface Area of Truncated Icosahedron given Icosahedral Edge Length Calculator

Formula Used:

\[ TSA = \frac{{l_{e(Icosahedron)}^2}}{3} \times \left( (10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})} \right) \]

Total Surface Area (TSA):

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1. What is the Total Surface Area of a 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 formula:

\[ TSA = \frac{{l_{e(Icosahedron)}^2}}{3} \times \left( (10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})} \right) \]

Where:

\( TSA \) — Total Surface Area of Truncated Icosahedron (m²)
\( l_{e(Icosahedron)} \) — Icosahedral Edge Length of Truncated Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the edge length of the original icosahedron from which the truncated icosahedron is derived.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is fundamental in various fields including architecture, materials science, and 3D modeling. For truncated icosahedrons (which resemble soccer balls), this calculation has practical applications in sports equipment design and molecular structures.

4. Using the Calculator

Tips: Enter the icosahedral edge length 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, resulting in 32 faces (12 pentagons and 20 hexagons).

Q2: Why is this shape significant?
A: This shape is famous as the pattern used in soccer balls and also appears in molecular structures like fullerenes (buckyballs).

Q3: How accurate is this formula?
A: The formula provides exact mathematical results for perfect geometric solids.

Q4: Can this calculator handle different units?
A: The calculator uses meters as input, but you can convert from other units before entering the value.

Q5: What are practical applications of this calculation?
A: Applications include material science research, architectural design, sports equipment manufacturing, and molecular modeling.