Total Surface Area of Truncated Icosahedron given Volume Calculator

Formula Used:

\[ TSA = 3 \times \left( \frac{4 \times V}{125 + 43 \times \sqrt{5}} \right)^{\frac{2}{3}} \times \left( 10 \times \sqrt{3} + \sqrt{25 + 10 \times \sqrt{5}} \right) \]

Total Surface Area of Truncated Icosahedron:

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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 formula:

\[ TSA = 3 \times \left( \frac{4 \times V}{125 + 43 \times \sqrt{5}} \right)^{\frac{2}{3}} \times \left( 10 \times \sqrt{3} + \sqrt{25 + 10 \times \sqrt{5}} \right) \]

Where:

\( TSA \) — Total Surface Area of Truncated Icosahedron (m²)
\( V \) — Volume of Truncated Icosahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the surface area from the volume of a truncated icosahedron using mathematical relationships between these geometric properties.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material science, architecture, and engineering design where the truncated icosahedron shape is used, such as in soccer balls and certain molecular structures.

4. Using the Calculator

Tips: Enter the volume of the truncated icosahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

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 for its appearance in soccer balls and its occurrence in nature, particularly in carbon fullerene molecules (buckyballs).

Q3: What are the units for the calculation?
A: The calculator uses cubic meters for volume and square meters for surface area. Ensure consistent units for accurate results.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated icosahedra. Other polyhedra have different mathematical relationships between volume and surface area.

Q5: What if I have the edge length instead of volume?
A: Different formulas exist for calculating surface area directly from edge length. This calculator specifically converts volume to surface area.