Midsphere Radius of Cuboctahedron given Volume Calculator

Formula Used:

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

Midsphere Radius of Cuboctahedron (rₘ):

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1. What is Midsphere Radius of Cuboctahedron?

The Midsphere Radius of a Cuboctahedron is the radius of the sphere that is tangent to every edge of the Cuboctahedron. This sphere lies between the insphere (tangent to faces) and the circumsphere (passing through all vertices) of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_m \) — Midsphere Radius of Cuboctahedron (m)
\( V \) — Volume of Cuboctahedron (m³)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{2} \) — Square root of 2 (approximately 1.414)

Explanation: This formula derives from the geometric properties of the cuboctahedron, relating its midsphere radius to its volume through a cubic root relationship.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and materials science for understanding the spatial properties of cuboctahedral structures, which appear in various natural and synthetic materials.

4. Using the Calculator

Tips: Enter the volume of the cuboctahedron in cubic meters. The volume must be a positive value 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 midsphere different from insphere and circumsphere?
A: The insphere is tangent to all faces, the circumsphere passes through all vertices, while the midsphere is tangent to all edges of the polyhedron.

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

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

Q5: What is the range of valid volume values?
A: The volume must be greater than zero. There's no theoretical upper limit, but extremely large values may exceed computational precision.