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Surface To Volume Ratio Of Snub Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ RA/V = \frac{(20\sqrt{3}+3\sqrt{25+10\sqrt{5}})\cdot6(3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}}}{\sqrt{\frac{TSA}{20\sqrt{3}+3\sqrt{25+10\sqrt{5}}}}\cdot((12(3\phi+1)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2-((36\phi+7)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}))-(53\phi+6))} \]

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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 measure that relates the total surface area of this complex polyhedron to its volume. It's an important geometric property that helps understand how the shape scales and its efficiency in various applications.

2. How Does the Calculator Work?

The calculator uses the complex mathematical formula:

\[ RA/V = \frac{(20\sqrt{3}+3\sqrt{25+10\sqrt{5}})\cdot6(3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}}}{\sqrt{\frac{TSA}{20\sqrt{3}+3\sqrt{25+10\sqrt{5}}}}\cdot((12(3\phi+1)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2-((36\phi+7)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}))-(53\phi+6))} \]

Where:

Explanation: This complex formula accounts for the intricate geometry of the snub dodecahedron, one of the Archimedean solids with 92 faces.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in materials science, chemistry, and physics as it affects properties like diffusion rates, heat transfer, and chemical reactivity. For polyhedra, it helps understand their geometric efficiency.

4. Using the Calculator

Tips: Enter the total surface area of the snub dodecahedron in square meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is the formula so complex?
A: The complexity arises from the irregular nature of the snub dodecahedron, which doesn't have the full symmetry of a regular dodecahedron.

Q3: What are typical values for surface to volume ratio?
A: The ratio depends on the size of the polyhedron. For a snub dodecahedron with TSA of 5500 m², the ratio is approximately 0.147 m⁻¹.

Q4: What practical applications does this calculation have?
A: This calculation is used in crystallography, nanotechnology, architecture, and any field dealing with complex polyhedral structures.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect snub dodecahedron, though real-world measurements may introduce some error.

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