1. What is the 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 vertices lie on the sphere's surface. It represents the smallest sphere that can completely enclose the truncated cuboctahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumsphere radius (m)
• \( TSA \) — Total surface area of truncated cuboctahedron (m²)
• \( \sqrt{} \) — Square root function

Explanation: This formula derives from the geometric properties of the truncated cuboctahedron, relating the circumsphere radius to the total surface area through mathematical constants and relationships specific to this Archimedean solid.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometry, crystallography, and materials science for understanding the spatial dimensions and packing efficiency of truncated cuboctahedron structures. It helps in determining the minimum container size needed to enclose such polyhedra.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero. The calculator will compute the circumsphere radius based on the mathematical relationship specific to truncated cuboctahedrons.

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 formula so complex?
A: The complexity arises from the intricate geometry of the truncated cuboctahedron, which requires mathematical constants like √2 and √3 to accurately describe its proportions and relationships.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the size of the polyhedron. For a standard truncated cuboctahedron with edge length 1, the circumsphere radius is approximately 2.847 units.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated cuboctahedrons. Other polyhedra have different mathematical relationships between surface area and circumsphere radius.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for ideal truncated cuboctahedrons. The accuracy depends on the precision of the input surface area measurement.