Home Back

Surface To Volume Ratio Of Snub Dodecahedron Given Midsphere Radius Calculator

Surface To Volume Ratio Of Snub Dodecahedron Formula:

\[ \text{Surface to Volume Ratio} = \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{(2 \times \text{Midsphere Radius})/\sqrt{\frac{1}{1-0.94315125924}} \times (((12((3\phi)+1))(((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

m

Unit Converter ▲

Unit Converter ▼

From: To:

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 of this complex polyhedron to its volume. 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 mathematical formula:

\[ \text{Surface to Volume Ratio} = \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{(2 \times \text{Midsphere Radius})/\sqrt{\frac{1}{1-0.94315125924}} \times (((12((3\phi)+1))(((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

Where:

Explanation: This complex formula accounts for the intricate geometry of the Snub Dodecahedron, one of the Archimedean solids, incorporating the golden ratio which is fundamental to its structure.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various 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-dependent phenomena.

4. Using the Calculator

Tips: Enter the midsphere radius of the Snub Dodecahedron in meters. The value must be positive and non-zero. The calculator will compute the surface to volume ratio using the precise mathematical formula.

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 (φ) used in this formula?
A: The golden ratio appears naturally in the geometry of the Snub Dodecahedron, making it fundamental to calculations involving this polyhedron.

Q3: What is the midsphere radius?
A: The midsphere radius is the radius of a sphere that is tangent to all edges of the polyhedron.

Q4: 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.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Snub Dodecahedron. Other polyhedra have different formulas for their surface to volume ratios.

Surface To Volume Ratio Of Snub Dodecahedron Given Midsphere Radius Calculator© - All Rights Reserved 2025