Home Back

Circumsphere Radius Of Snub Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{\frac{2-0.94315125924}{1-0.94315125924}}}{2} \times \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{RA/V \times (((12((3\phi)+1))(((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

m⁻¹

Unit Converter ▲

Unit Converter ▼

From: To:

1. What is Circumsphere Radius of Snub Dodecahedron?

The circumsphere radius of a snub dodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. It's a fundamental geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the complex mathematical formula:

\[ r_c = \frac{\sqrt{\frac{2-0.94315125924}{1-0.94315125924}}}{2} \times \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{RA/V \times (((12((3\phi)+1))(((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7)((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

Where:

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions and geometric properties of the snub dodecahedron, which has applications in crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in m⁻¹. The value must be positive and non-zero. The calculator will compute the corresponding circumsphere radius.

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 the formula?
A: The golden ratio appears naturally in the geometry of regular pentagons and dodecahedrons, making it fundamental to calculations involving these shapes.

Q3: What are typical values for surface to volume ratio?
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 be used for other polyhedra?
A: No, this specific formula is designed only for the snub dodecahedron. Other polyhedra have different geometric relationships.

Q5: How accurate is the calculation?
A: The calculation uses precise mathematical constants and should be highly accurate, though rounding may occur in the final result display.

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