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Surface To Volume Ratio Of Snub Dodecahedron Given Circumsphere Radius Calculator

Surface to Volume Ratio of Snub Dodecahedron Formula:

\[ \frac{A}{V} = \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{(2 \times r_c)/\sqrt{\frac{2-0.94315125924}{1-0.94315125924}} \times (((12 \times ((3\phi)+1)) \times (((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7) \times ((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

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1. What is Surface to Volume Ratio of Snub Dodecahedron?

The Surface to Volume Ratio of a Snub Dodecahedron is a geometric measurement that compares the total surface area to the volume of this complex polyhedron. It's an important parameter in materials science, chemistry, and physics where surface properties relative to volume are significant.

2. How Does the Calculator Work?

The calculator uses the complex formula:

\[ \frac{A}{V} = \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{(2 \times r_c)/\sqrt{\frac{2-0.94315125924}{1-0.94315125924}} \times (((12 \times ((3\phi)+1)) \times (((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7) \times ((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

Where:

Explanation: This complex formula accounts for the intricate geometry of the snub dodecahedron, one of the Archimedean solids, using the golden ratio and cube roots to precisely calculate the surface to volume ratio.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in many scientific fields. In materials science, it affects reactivity and strength. In chemistry, it influences reaction rates. In physics, it relates to heat transfer and other surface phenomena.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive. The calculator will compute the surface to volume ratio using the precise mathematical formula for a snub dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a snub dodecahedron?
A: A snub dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles), 150 edges, and 60 vertices.

Q2: Why is the formula so complex?
A: The snub dodecahedron has a highly irregular structure that requires complex mathematical expressions involving the golden ratio and cube roots to accurately describe its geometry.

Q3: What are typical values for surface to volume ratio?
A: The ratio depends on the size of the polyhedron. Smaller snub dodecahedra have higher surface to volume ratios than larger ones of the same shape.

Q4: What units does this calculator use?
A: The calculator uses meters for input (circumsphere radius) and returns m⁻¹ (inverse meters) for the surface to volume ratio.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula is only valid for snub dodecahedra. Other polyhedra have different formulas for calculating surface to volume ratio.

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