Volume of Truncated Icosahedron given Total Surface Area Calculator

Formula Used:

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

Volume of Truncated Icosahedron:

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1. What is the Volume of Truncated Icosahedron?

The truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons). It is best known as the shape of a soccer ball. The volume calculation is essential in geometry and various engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of the truncated icosahedron
\( TSA \) — Total surface area
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula derives the volume from the total surface area by first calculating the edge length and then applying the volume formula for a truncated icosahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, physics, engineering, and architecture. For the truncated icosahedron, this is particularly relevant in materials science, chemical structures (fullerenes), and sports equipment design.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive. The calculator will compute the corresponding volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid formed by truncating the vertices of an icosahedron, resulting in 12 pentagonal faces and 20 hexagonal faces.

Q2: Where is this shape commonly found?
A: The most familiar example is a soccer ball. It's also found in molecular structures like buckminsterfullerene (C₆₀).

Q3: What are the units for input and output?
A: Input is in square meters (m²) for surface area, output is in cubic meters (m³) for volume. Consistent units must be used.

Q4: Can this formula be used for any truncated icosahedron?
A: Yes, this formula applies to all regular truncated icosahedrons where all edges are of equal length.

Q5: What if I have the edge length instead of surface area?
A: If you have the edge length (a), the volume can be calculated more directly using: \( V = \frac{125 + 43\sqrt{5}}{4} a^3 \)