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Volume Of Snub Dodecahedron Given Midsphere Radius Calculator

Formula Used:

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

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1. What is the Volume of Snub Dodecahedron?

The Snub Dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles). Its volume represents the total three-dimensional space enclosed by its surface, calculated based on its midsphere radius.

2. How Does the Calculator Work?

The calculator uses the complex mathematical formula:

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

Where:

Explanation: This complex formula accounts for the unique geometric properties of the Snub Dodecahedron, incorporating the golden ratio and various mathematical operations to accurately calculate its volume.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and architecture. For the Snub Dodecahedron, volume calculation helps in understanding its spatial properties, material requirements, and applications in various fields including crystallography and structural design.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the volume using the complex mathematical formula shown above.

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

Q3: Why is the golden ratio used in the formula?
A: The golden ratio appears naturally in the geometry of many polyhedra, including the Snub Dodecahedron, due to its unique mathematical properties.

Q4: What are practical applications of this calculation?
A: This calculation is used in mathematical research, architectural design, material science, and in understanding the properties of complex geometric structures.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise, using the exact formula derived from the geometric properties of the Snub Dodecahedron.

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