Midsphere Radius of Truncated Cuboctahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ r_m = \frac{\sqrt{12 + 6\sqrt{2}}}{2} \times \frac{6(2 + \sqrt{2} + \sqrt{3})}{RA/V \times (11 + 7\sqrt{2})} \]

Midsphere Radius (rₘ):

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1. What is Midsphere Radius of Truncated Cuboctahedron?

The Midsphere Radius of a Truncated Cuboctahedron is the radius of the sphere that is tangent to all edges of the polyhedron. It represents the sphere that fits perfectly within the polyhedron, touching each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{\sqrt{12 + 6\sqrt{2}}}{2} \times \frac{6(2 + \sqrt{2} + \sqrt{3})}{RA/V \times (11 + 7\sqrt{2})} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{2} \) — Square root of 2 (≈1.4142)
\( \sqrt{3} \) — Square root of 3 (≈1.7321)

Explanation: This formula calculates the midsphere radius based on the surface to volume ratio of a truncated cuboctahedron, incorporating geometric constants specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: The midsphere radius is important in geometry and materials science for understanding the spatial properties and packing efficiency of polyhedral structures. It helps in analyzing the geometric relationships between different dimensions of the polyhedron.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for valid 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 regular hexagons, and 6 regular octagons.

Q2: How is surface to volume ratio defined?
A: Surface to volume ratio is the total surface area of the polyhedron divided by its volume, measured in 1/m.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the specific dimensions of the polyhedron and can vary significantly based on the surface to volume ratio.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the base unit. Ensure consistent units when inputting values.

Q5: What applications use midsphere radius calculations?
A: These calculations are used in crystallography, nanotechnology, materials science, and geometric modeling applications.