Surface to Volume Ratio of Hexakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ RA/V = \frac{6}{5} \times \sqrt{\frac{10 \times (417 + 107 \times \sqrt{5})}{6 \times (185 + 82 \times \sqrt{5})}} \times \frac{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}}{4 \times r_i} \]

Surface to Volume Ratio of Hexakis Icosahedron:

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

The Surface to Volume Ratio of Hexakis Icosahedron is a geometric property that represents what part of or fraction of total volume of Hexakis Icosahedron is the total surface area. It's an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

\( RA/V \) — Surface to Volume Ratio (m⁻¹)
\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the surface to volume ratio based on the insphere radius of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and engineering. It helps in understanding properties like heat transfer, reaction rates, and structural efficiency of complex geometric shapes.

4. Using the Calculator

Tips: Enter the insphere radius of the Hexakis Icosahedron in 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 does the surface to volume ratio indicate?
A: It indicates how much surface area is available per unit volume, which is important for processes involving surface interactions.

Q3: What are typical values for this ratio?
A: The ratio depends on the insphere radius. Smaller radii generally yield higher surface to volume ratios.

Q4: Can this calculator handle very small or very large values?
A: Yes, as long as the input values are positive and within reasonable computational limits.

Q5: What units should I use for the insphere radius?
A: The calculator uses meters, but you can use any consistent unit system as long as you interpret the results accordingly.