Volume of Truncated Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ V = \frac{125 + 43\sqrt{5}}{4} \times \left( \frac{12 \times (10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{RA/V \times (125 + 43\sqrt{5})} \right)^3 \]

Volume of Truncated Icosahedron:

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1. What is the Volume of Truncated Icosahedron?

The volume of a truncated icosahedron is the total quantity of three dimensional space enclosed by the surface of this Archimedean solid. A truncated icosahedron is a polyhedron with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{125 + 43\sqrt{5}}{4} \times \left( \frac{12 \times (10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{RA/V \times (125 + 43\sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of the truncated icosahedron (m³)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the volume of a truncated icosahedron based on its surface to volume ratio, using the mathematical properties of this specific polyhedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in various fields including architecture, engineering, materials science, and mathematics. For truncated icosahedrons specifically, this calculation is important in understanding the properties of fullerene molecules and other applications in nanotechnology.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/m. The value must be positive and greater than zero. The calculator will compute the corresponding volume of the truncated icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons), 90 edges, and 60 vertices. It's the shape of a soccer ball.

Q2: What are real-world applications of truncated icosahedrons?
A: This shape appears in fullerene molecules (C60 buckyballs), soccer balls, and various architectural and engineering designs.

Q3: How is surface to volume ratio related to volume?
A: The surface to volume ratio is inversely proportional to the size of the object. For a given shape, as volume increases, the surface to volume ratio decreases.

Q4: What units should I use for the calculation?
A: Use consistent units. If surface to volume ratio is in 1/meter, the volume result will be in cubic meters.

Q5: Are there limitations to this calculation?
A: This formula assumes a perfect geometric truncated icosahedron. Real-world objects may have manufacturing tolerances or imperfections.