Circumsphere Radius of Rhombicuboctahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{5 + 2\sqrt{2}}}{2} \times \frac{3(9 + \sqrt{3})}{R_{A/V} \times (6 + 5\sqrt{2})} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Rhombicuboctahedron?

The Circumsphere Radius of a Rhombicuboctahedron is the radius of the sphere that contains the polyhedron in such a way that all the vertices are lying on the sphere. It represents the distance from the center of the polyhedron to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{5 + 2\sqrt{2}}}{2} \times \frac{3(9 + \sqrt{3})}{R_{A/V} \times (6 + 5\sqrt{2})} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( R_{A/V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{2} \) — Square root of 2 (≈1.4142)
\( \sqrt{3} \) — Square root of 3 (≈1.7321)

Explanation: This formula calculates the circumsphere radius based on the surface to volume ratio of the rhombicuboctahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Circumsphere Radius Calculation

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

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and greater than zero for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombicuboctahedron?
A: A rhombicuboctahedron is an Archimedean solid with 8 triangular and 18 square faces, 24 identical vertices, and 48 edges.

Q2: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size of the polyhedron. Smaller structures have higher ratios, while larger ones have lower ratios.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the rhombicuboctahedron only. Other polyhedra have different geometric relationships.

Q4: What are the units of measurement?
A: The circumsphere radius is measured in meters (m) and surface to volume ratio in reciprocal meters (m⁻¹).

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the ideal rhombicuboctahedron.