Circumsphere Radius of Icosidodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{(1+\sqrt{5}) \times 3 \times (5\sqrt{3} + 3\sqrt{25+10\sqrt{5}})}{(RA/V) \times (45+17\sqrt{5})} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Icosidodecahedron?

The Circumsphere Radius of an Icosidodecahedron is the radius of the sphere that contains the Icosidodecahedron in such a way that all the vertices are lying on the sphere. It is an important geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{(1+\sqrt{5}) \times 3 \times (5\sqrt{3} + 3\sqrt{25+10\sqrt{5}})}{(RA/V) \times (45+17\sqrt{5})} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the circumsphere radius based on the surface to volume ratio of the icosidodecahedron, incorporating mathematical constants and geometric 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 icosidodecahedral structures.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and valid for accurate calculation of the circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is an Icosidodecahedron?
A: An Icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 30 vertices, and 60 edges.

Q2: What is the significance of the circumsphere?
A: The circumsphere is the smallest sphere that can contain the polyhedron, touching all its vertices, which is important for spatial analysis.

Q3: How is surface to volume ratio related to circumsphere radius?
A: The surface to volume ratio affects the spatial dimensions of the polyhedron, which in turn determines the size of its circumsphere.

Q4: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the specific dimensions of the icosidodecahedron and can vary based on its surface to volume ratio.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the icosidodecahedron. Other polyhedra have different geometric relationships.