1. What is Circumsphere Radius of Cuboctahedron?

The Circumsphere Radius of a Cuboctahedron is the radius of the sphere that contains the Cuboctahedron in such a way that all the vertices are lying on the sphere. It's a fundamental geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumsphere Radius (m)
• \( RA/V \) — Surface to Volume Ratio (1/m)
• \( \sqrt{3} \) — Square root of 3 (approximately 1.732)
• \( \sqrt{2} \) — Square root of 2 (approximately 1.414)

Explanation: This formula relates the circumsphere radius of a cuboctahedron to its surface-to-volume ratio through a precise mathematical relationship.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, materials science, and crystallography where cuboctahedral structures appear. It helps in understanding the spatial dimensions and packing efficiency of these structures.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Cuboctahedron?
A: A cuboctahedron is an Archimedean solid with 8 triangular faces and 6 square faces, having 12 identical vertices and 24 identical edges.

Q2: What are typical values for Surface to Volume Ratio?
A: The surface-to-volume ratio depends on the size of the cuboctahedron. 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 applies only to cuboctahedra. Other polyhedra have different relationships between circumsphere radius and surface-to-volume ratio.

Q4: What units should I use?
A: Use consistent units. If surface-to-volume ratio is in 1/meters, the circumsphere radius will be in meters.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect cuboctahedron, assuming precise input values.