Volume Of Cuboctahedron Given Midsphere Radius Calculator

Cuboctahedron Volume Formula:

\[ V = \frac{5\sqrt{2}}{3} \left( \frac{2}{\sqrt{3}} \cdot r_m \right)^3 \]

Volume of Cuboctahedron (V):

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1. What is the Cuboctahedron Volume Formula?

The cuboctahedron volume formula calculates the 3-dimensional space enclosed by the surface of a cuboctahedron given its midsphere radius. A cuboctahedron is an Archimedean solid with 8 triangular faces and 6 square faces.

2. How Does the Calculator Work?

The calculator uses the cuboctahedron volume formula:

\[ V = \frac{5\sqrt{2}}{3} \left( \frac{2}{\sqrt{3}} \cdot r_m \right)^3 \]

Where:

\( V \) — Volume of the cuboctahedron (m³)
\( r_m \) — Midsphere radius (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{3} \) — Square root of 3 (approximately 1.7321)

Explanation: The formula derives from the geometric properties of the cuboctahedron, relating its volume to the radius of the sphere tangent to all its edges.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and material science. It helps in understanding spatial relationships and material requirements.

4. Using the Calculator

Tips: Enter the midsphere radius in 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 14 faces (8 equilateral triangles and 6 squares), 12 identical vertices, and 24 edges.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to every edge of the cuboctahedron, located between its insphere and circumsphere.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to cuboctahedra. Different polyhedra have different volume formulas based on their unique geometric properties.

Q4: What are practical applications of cuboctahedron volume calculation?
A: Applications include crystallography (crystal structures), architectural design, molecular modeling, and geometric optimization problems.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula. The accuracy depends on the precision of the input midsphere radius value.