Circumsphere Radius of Snub Dodecahedron given Total Surface Area Calculator

Formula Used:

\[ r_c = \frac{\sqrt{\frac{2-0.94315125924}{1-0.94315125924}}}{2} \times \sqrt{\frac{TSA}{(20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})}} \]

Circumsphere Radius of Snub Dodecahedron:

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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 in such a way that all the vertices are lying on the sphere. It's an important geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{\frac{2-0.94315125924}{1-0.94315125924}}}{2} \times \sqrt{\frac{TSA}{(20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( TSA \) — Total Surface Area of Snub Dodecahedron (m²)
0.94315125924 — A constant specific to the Snub Dodecahedron

Explanation: The formula calculates the circumsphere radius based on the total surface area of the polyhedron, using specific geometric constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, crystallography, and materials science for understanding the spatial properties and packing efficiency of polyhedral structures.

4. Using the Calculator

Tips: Enter the total surface area of the Snub Dodecahedron in square meters. The value must be positive and greater than zero.

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 are typical values for circumsphere radius?
A: The circumsphere radius depends on the size of the polyhedron. For a standard Snub Dodecahedron with unit edge length, the circumsphere radius is approximately 2.1558 units.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Snub Dodecahedron, assuming precise input values.

Q4: 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 calculating circumsphere radius.

Q5: What units should I use?
A: Use consistent units. If you input surface area in m², the result will be in meters. The calculator works with any consistent unit system.