Circumsphere Radius of Truncated Cuboctahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{13 + 6\sqrt{2}}}{2} \times \frac{6(2 + \sqrt{2} + \sqrt{3})}{RA/V \times (11 + 7\sqrt{2})} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Truncated Cuboctahedron?

The Circumsphere Radius of a Truncated Cuboctahedron is the radius of the sphere that contains the polyhedron in such a way that all its vertices lie on the sphere's surface. It is an important geometric property in polyhedral geometry.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{13 + 6\sqrt{2}}}{2} \times \frac{6(2 + \sqrt{2} + \sqrt{3})}{RA/V \times (11 + 7\sqrt{2})} \]

Where:

\( r_c \) — Circumsphere radius (meters)
\( RA/V \) — Surface to Volume Ratio (1/meter)
\( \sqrt{2} \) — 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 a truncated cuboctahedron, incorporating various mathematical constants and geometric relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial dimensions and geometric properties of truncated cuboctahedrons, which have applications in crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/meter units. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated cuboctahedron?
A: A truncated cuboctahedron is an Archimedean solid with 26 faces (12 squares, 8 regular hexagons, and 6 regular octagons), 72 edges, and 48 vertices.

Q2: Why is the circumsphere radius important?
A: The circumsphere radius helps determine the smallest sphere that can contain the polyhedron, which is useful for packaging, spatial analysis, and geometric computations.

Q3: What units should I use for surface to volume ratio?
A: Use 1/meter units for consistency with the formula. Make sure your input value matches this unit system.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect truncated cuboctahedron shape and may not account for manufacturing tolerances or material properties in real-world applications.

Q5: Can this formula be used for other polyhedrons?
A: No, this specific formula is designed only for truncated cuboctahedrons. Other polyhedrons have different geometric relationships and require separate formulas.