Circumsphere Radius of Truncated Cuboctahedron given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{13 + 6\sqrt{2}}}{2} \times \left( \frac{V}{2(11 + 7\sqrt{2})} \right)^{\frac{1}{3}} \]

Circumsphere Radius (r_c):

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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:

\[ r_c = \frac{\sqrt{13 + 6\sqrt{2}}}{2} \times \left( \frac{V}{2(11 + 7\sqrt{2})} \right)^{\frac{1}{3}} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of truncated cuboctahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula derives from the geometric properties of the truncated cuboctahedron, relating its circumsphere radius to its volume through mathematical constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, material science, and architectural design for understanding the spatial requirements and packaging efficiency of this complex polyhedral shape.

4. Using the Calculator

Tips: Enter the volume of the truncated cuboctahedron in cubic meters. 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 hexagons, and 6 octagons), 72 edges, and 48 vertices.

Q2: How is this formula derived?
A: The formula is derived from the geometric relationships between the volume and circumsphere radius of the truncated cuboctahedron, using mathematical constants specific to this polyhedron.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the volume. For a unit volume truncated cuboctahedron, the circumsphere radius is approximately 0.92 units.

Q4: Can this calculator handle different units?
A: The calculator uses cubic meters for volume and meters for radius. Convert other units to meters before calculation for accurate results.

Q5: What are practical applications of this calculation?
A: This calculation is useful in crystallography, molecular modeling, architectural design, and any field dealing with complex polyhedral structures.