Surface to Volume Ratio of Hexakis Icosahedron given Volume Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{6}{5} \times \sqrt{\frac{10 \times (417 + 107 \times \sqrt{5})}{6 \times (185 + 82 \times \sqrt{5})}} \times \left( \frac{25 \times \sqrt{6 \times (185 + 82 \times \sqrt{5})}}{88 \times V} \right)^{\frac{1}{3}} \]

Surface to Volume Ratio:

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

The Surface to Volume Ratio of a Hexakis Icosahedron represents what part of or fraction of total volume is the total surface area. It's an important geometric property that indicates how much surface area is available per unit volume of the shape.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( V \) — Volume of Hexakis Icosahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the ratio by considering the complex geometric properties of the Hexakis Icosahedron, including its surface area and volume relationships.

3. Importance of Surface to Volume Ratio

Details: This ratio is crucial in various fields including materials science, chemistry, and engineering where surface interactions relative to volume are important, such as in catalysis, heat transfer, and fluid dynamics applications.

4. Using the Calculator

Tips: Enter the volume of the Hexakis 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 Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: What units should I use for volume?
A: The calculator expects volume in cubic meters (m³). If you have measurements in other units, convert them to cubic meters first.

Q3: Can this calculator handle very small or very large volumes?
A: Yes, the calculator can handle a wide range of volume values, but extremely small values near zero may cause mathematical issues.

Q4: What does the surface to volume ratio tell us about the shape?
A: A higher ratio indicates more surface area relative to volume, which is important in processes where surface interactions dominate.

Q5: Are there practical applications for this calculation?
A: Yes, this calculation is useful in nanotechnology, materials design, pharmaceutical research, and any field where the relationship between surface area and volume affects material properties.