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Midsphere Radius Of Snub Dodecahedron Given Volume Calculator

Midsphere Radius of Snub Dodecahedron Formula:

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

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1. What is Midsphere Radius of Snub Dodecahedron?

The midsphere radius of a snub dodecahedron is the radius of the sphere that is tangent to all edges of the polyhedron. It represents the sphere that fits perfectly between the inscribed and circumscribed spheres of the snub dodecahedron.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

Where:

Explanation: This complex formula relates the midsphere radius to the volume of the snub dodecahedron using the mathematical constant phi (the golden ratio).

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of the snub dodecahedron, which is one of the Archimedean solids with interesting mathematical properties.

4. Using the Calculator

Tips: Enter the volume of the snub dodecahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

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 accurate calculations of its properties.

Q3: What are the practical applications of this calculation?
A: This calculation is used in mathematical research, computer graphics, architectural design, and the study of polyhedral geometry.

Q4: How accurate is this calculator?
A: The calculator provides high precision results using the exact mathematical formula with the golden ratio constant.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the snub dodecahedron. Other polyhedra have different formulas for calculating their midsphere radii.

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