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Surface To Volume Ratio Of Snub Dodecahedron Given Volume Calculator

Surface To Volume Ratio Of Snub Dodecahedron Given Volume Formula:

\[ \text{Surface to Volume Ratio} = \frac{(20\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}) \times 6(3 - ((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}}}{\left(\frac{V \times 6(3 - ((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}}}{(12(3\phi + 1)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2 - (36\phi + 7)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}}) - (53\phi + 6)}\right)^{\frac{1}{3}} \times (12(3\phi + 1)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2 - (36\phi + 7)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}}) - (53\phi + 6)} \]

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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 compares the total surface area of this polyhedron to its volume. It's an important geometric property that indicates how much surface area is available per unit volume of the shape.

2. How Does the Calculator Work?

The calculator uses the complex mathematical formula:

\[ \text{Surface to Volume Ratio} = \frac{(20\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}) \times 6(3 - ((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}}}{\left(\frac{V \times 6(3 - ((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}}}{(12(3\phi + 1)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2 - (36\phi + 7)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}}) - (53\phi + 6)}\right)^{\frac{1}{3}} \times (12(3\phi + 1)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}})^2 - (36\phi + 7)((\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}} + (\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2})^{\frac{1}{3}}) - (53\phi + 6)} \]

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3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and physics. For polyhedra like the snub dodecahedron, this ratio helps understand properties related to surface interactions, heat transfer, and other physical characteristics that depend on the relationship between surface area and volume.

4. Using the Calculator

Tips: Enter the volume of the snub dodecahedron in cubic meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio using the complex mathematical formula involving the golden ratio.

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 golden ratio used in this calculation?
A: The golden ratio appears naturally in the geometry of the snub dodecahedron and is fundamental to its mathematical description.

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

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

Q5: What units should I use for volume?
A: The calculator expects volume in cubic meters, and returns surface to volume ratio in meters⁻¹ (inverse meters).

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