Circumsphere Radius Of Cuboctahedron Given Volume Calculator

Formula Used:

\[ r_c = \left( \frac{3 \times V}{5 \times \sqrt{2}} \right)^{\frac{1}{3}} \]

Circumsphere Radius (r_c):

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1. What is the 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 represents the distance from the center of the cuboctahedron to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \left( \frac{3 \times V}{5 \times \sqrt{2}} \right)^{\frac{1}{3}} \]

Where:

\( r_c \) — Circumsphere radius of cuboctahedron (meters)
\( V \) — Volume of cuboctahedron (cubic meters)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the circumsphere radius based on the volume of the cuboctahedron, using the mathematical relationship between these geometric properties.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, material science, and structural engineering for understanding the spatial dimensions and packing efficiency of cuboctahedral structures.

4. Using the Calculator

Tips: Enter the volume of the 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 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: How is the circumsphere different from the insphere?
A: The circumsphere passes through all vertices of the polyhedron, while the insphere is tangent to all faces of the polyhedron.

Q3: What are typical applications of cuboctahedrons?
A: Cuboctahedral structures are found in crystallography, nanotechnology, and architectural design due to their efficient space-filling properties.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to cuboctahedrons. Other polyhedrons have different relationships between volume and circumsphere radius.

Q5: What units should be used for the volume input?
A: The calculator expects volume in cubic meters, but you can use any consistent unit system as long as the circumsphere radius will be in the corresponding linear units.