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Circumsphere Radius Of Snub Dodecahedron Given Volume Calculator

Circumsphere Radius of Snub Dodecahedron Formula:

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

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1. What is the Circumsphere Radius of Snub Dodecahedron?

The circumsphere radius of a snub dodecahedron is the radius of the sphere that contains the polyhedron in such a way 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 formula:

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

Where:

Explanation: This complex formula relates the circumsphere radius to the volume through the mathematical properties of the snub dodecahedron and the golden ratio.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, crystallography, and materials science for understanding the spatial properties of this complex polyhedron and its applications in various fields.

4. Using the Calculator

Tips: Enter the volume of the snub dodecahedron in cubic meters. The volume must be a positive value. The calculator will compute the circumsphere radius using the complex mathematical relationship.

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 formula?
A: The golden ratio appears naturally in the geometry of the snub dodecahedron, making it essential for calculating its various properties.

Q3: What are practical applications of this calculation?
A: This calculation is used in molecular modeling, crystallography, architectural design, and the study of geometric properties of complex shapes.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect snub dodecahedron, though practical measurements may have some margin of error.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the snub dodecahedron. Other polyhedra have different mathematical relationships between volume and circumsphere radius.

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