Surface to Volume Ratio of Hexakis Icosahedron given Truncated Icosidodecahedron Edge Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{6}{5} \times \sqrt{\frac{10 \times (417 + 107 \times \sqrt{5})}{6 \times (185 + 82 \times \sqrt{5})}} \times \frac{5}{2 \times l_{e} \times \sqrt{15 \times (5 - \sqrt{5})}} \]

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 is a geometric property that represents the relationship between the total surface area and the total volume of this complex polyhedron. It 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:

\( l_{e} \) — Truncated Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula calculates the ratio of surface area to volume based on the truncated edge length of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is important in various fields including materials science, chemistry, and physics. For polyhedra, it helps understand properties like diffusion rates, heat transfer, and structural efficiency.

4. Using the Calculator

Tips: Enter the truncated edge length 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 icosahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: What are typical values for surface to volume ratio?
A: The ratio depends on the size of the polyhedron. Smaller polyhedra typically have higher surface to volume ratios than larger ones of the same shape.

Q3: How is this ratio used in practical applications?
A: In materials science, it helps predict how substances interact with their environment. In chemistry, it affects reaction rates for surface-mediated processes.

Q4: What are the units of surface to volume ratio?
A: The ratio is expressed in reciprocal length units (m⁻¹), as it represents area divided by volume (m²/m³ = m⁻¹).

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Hexakis Icosahedron with a truncated icosahedron edge as input.