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

Formula Used:

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

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1. What is Total Surface Area Of Snub Dodecahedron Given Surface To Volume Ratio?

The Total Surface Area of Snub Dodecahedron given Surface to Volume Ratio is the total quantity of plane enclosed by the entire surface of the Snub Dodecahedron, calculated using its surface to volume ratio. This complex geometric calculation involves the golden ratio and various mathematical operations.

2. How Does the Calculator Work?

The calculator uses the complex formula:

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

Where:

Explanation: The formula accounts for the complex geometric properties of the snub dodecahedron, incorporating the golden ratio and various mathematical operations including square roots and cube roots.

3. Importance of TSA Calculation

Details: Calculating the total surface area of complex polyhedra like the snub dodecahedron is crucial for geometric analysis, material science applications, and understanding the properties of complex three-dimensional structures.

4. Using the Calculator

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

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 golden ratio used in this calculation?
A: The golden ratio appears naturally in the geometry of regular and semi-regular polyhedra, including the snub dodecahedron.

Q3: What are typical surface to volume ratio values for a snub dodecahedron?
A: The surface to volume ratio depends on the size of the polyhedron. For a unit snub dodecahedron, it's approximately 2.097.

Q4: What units should I use for input and output?
A: Input surface to volume ratio in m⁻¹, and the result will be in m². Ensure consistent units throughout your calculations.

Q5: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric shape and may not account for manufacturing tolerances or material properties in real-world applications.

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