Surface to Volume Ratio of Hexakis Icosahedron Calculator

Surface to Volume Ratio Formula:

\[ \text{Surface to Volume Ratio} = \frac{6}{5} \times \sqrt{\frac{10 \times (417 + 107 \times \sqrt{5})}{6 \times (185 + 82 \times \sqrt{5})}} \times \frac{1}{\text{Long Edge}} \]

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 the relationship between its total surface area and its total volume. It's an important geometric property that indicates how much surface area is available per unit volume of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{6}{5} \times \sqrt{\frac{10 \times (417 + 107 \times \sqrt{5})}{6 \times (185 + 82 \times \sqrt{5})}} \times \frac{1}{\text{Long Edge}} \]

Where:

Long Edge - The length of the longest edge of the Hexakis Icosahedron (in meters)
√5 - Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the ratio of surface area to volume based on the geometric properties of the Hexakis Icosahedron, which is a Catalan solid derived from the icosahedron.

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 affects properties like heat transfer, chemical reactivity, and mechanical strength. For polyhedra, this ratio helps understand their geometric efficiency and spatial properties.

4. Using the Calculator

Tips: Enter the length of the long edge of the Hexakis Icosahedron 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 with 120 faces, 180 edges, and 62 vertices. It's the dual polyhedron of the truncated dodecahedron.

Q2: Why is the surface to volume ratio important?
A: This ratio is important in many scientific applications, including determining reaction rates, heat dissipation, and understanding the geometric efficiency of shapes.

Q3: What units are used for the calculation?
A: The long edge should be entered in meters, and the result is given in reciprocal meters (m⁻¹).

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

Q5: What if I have the short edge measurement instead?
A: This calculator requires the long edge measurement. If you have the short edge, you'll need to convert it to the long edge using the appropriate geometric relationship for Hexakis Icosahedron.