Surface To Volume Ratio Of Hexakis Icosahedron Given Midsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{6}{5} \times \sqrt{\frac{10 \times (417 + 107 \times \sqrt{5})}{6 \times (185 + 82 \times \sqrt{5})}} \times \frac{5 + 3 \times \sqrt{5}}{8 \times \text{Midsphere Radius}} \]

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 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:

Midsphere Radius - The radius of the sphere that touches all edges of the Hexakis Icosahedron (in meters)
\(\sqrt{5}\) - Square root of 5 (approximately 2.23607)

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

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and physics. It helps determine properties like reactivity, heat transfer efficiency, and diffusion rates in polyhedral structures.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio in reciprocal meters (m⁻¹).

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 is the significance of the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron, providing a fundamental geometric measurement.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Hexakis Icosahedron, assuming precise input values.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Icosahedron. Other polyhedra have different surface to volume ratio formulas.

Q5: What are typical values for surface to volume ratio?
A: The ratio depends on the size of the polyhedron. Smaller polyhedra generally have higher surface to volume ratios than larger ones of the same shape.