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Edge Length of Snub Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

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

m⁻¹

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1. What is Edge Length of Snub Dodecahedron?

The edge length of a snub dodecahedron is the length of any edge of this Archimedean solid. The snub dodecahedron is a polyhedron with 92 faces (12 pentagons and 80 triangles), 150 edges, and 60 vertices.

2. How Does the Calculator Work?

The calculator uses the complex mathematical formula:

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

Where:

Explanation: This formula relates the edge length to the surface-to-volume ratio through complex mathematical relationships involving the golden ratio and various roots.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from surface-to-volume ratio is important in geometric modeling, material science, and architectural design where the snub dodecahedron's properties need to be precisely determined.

4. Using the Calculator

Tips: Enter the surface-to-volume ratio in m⁻¹. The value must be positive and valid for meaningful results.

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: What is the golden ratio (φ)?
A: The golden ratio is a mathematical constant approximately equal to 1.618034, often found in geometry, art, and nature.

Q3: What are typical surface-to-volume ratio values?
A: The surface-to-volume ratio depends on the size of the polyhedron. Smaller polyhedra have higher ratios, while larger ones have lower ratios.

Q4: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of values, but extreme values may result in computational limitations or precision issues.

Q5: What units should I use?
A: Use consistent units. If surface-to-volume ratio is in m⁻¹, the edge length result will be in meters.

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