Surface to Volume Ratio of Truncated Icosahedron given Midsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{9 \times (10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{\frac{r_m}{1 + \sqrt{5}} \times (125 + 43\sqrt{5})} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Truncated Icosahedron?

The Surface to Volume Ratio of a Truncated Icosahedron is a geometric property that represents the relationship between the total surface area and the volume of this polyhedron. It's particularly important in materials science and nanotechnology applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{9 \times (10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{\frac{r_m}{1 + \sqrt{5}} \times (125 + 43\sqrt{5})} \]

Where:

\( r_m \) — Midsphere radius of the truncated icosahedron
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the ratio of surface area to volume based on the midsphere radius, which is the radius of the sphere tangent to all edges of the polyhedron.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in determining various physical and chemical properties, including heat transfer efficiency, reaction rates, and mechanical strength in nanostructures and materials engineering.

4. Using the Calculator

Tips: Enter the midsphere radius in 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 with 32 faces (12 pentagons and 20 hexagons), best known as the shape of a soccer ball.

Q2: Why is midsphere radius used in this calculation?
A: The midsphere radius provides a convenient geometric reference point that simplifies the calculation of surface to volume ratio for this specific polyhedron.

Q3: What are typical values for surface to volume ratio?
A: The ratio depends on the size of the polyhedron. Smaller structures have higher surface to volume ratios, which is particularly important in nanotechnology applications.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for truncated icosahedra. Other polyhedra have different surface to volume ratio formulas.

Q5: What units should I use for the midsphere radius?
A: The calculator uses meters, but any consistent unit system can be used as long as the surface to volume ratio units match accordingly.