Midsphere Radius of Truncated Cuboctahedron given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{\sqrt{12 + 6\sqrt{2}}}{2} \times \sqrt{\frac{TSA}{12(2 + \sqrt{2} + \sqrt{3})}} \]

Midsphere Radius (rₘ):

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1. What is the 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 touches every edge of the Truncated Cuboctahedron at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{\sqrt{12 + 6\sqrt{2}}}{2} \times \sqrt{\frac{TSA}{12(2 + \sqrt{2} + \sqrt{3})}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( TSA \) — Total Surface Area (m²)
\( \sqrt{2} \) — Square root of 2 (≈1.4142)
\( \sqrt{3} \) — Square root of 3 (≈1.7321)

Explanation: This formula calculates the midsphere radius based on the total surface area of the truncated cuboctahedron, using geometric relationships between the polyhedron's dimensions.

3. Importance of Midsphere Radius Calculation

Details: The midsphere radius is important in geometry and materials science for understanding the spatial properties of polyhedra, particularly in crystallography and molecular modeling where truncated cuboctahedra appear.

4. Using the Calculator

Tips: Enter the total surface area in square 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 regular hexagons, and 6 regular octagons.

Q2: How is the midsphere different from the insphere?
A: The midsphere touches all edges, while the insphere touches all faces. Not all polyhedra have both spheres.

Q3: What units should I use for the calculation?
A: Use consistent units (typically meters for length and square meters for area). The result will be in the same length unit as the input.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Truncated Cuboctahedron. Other polyhedra have different geometric relationships.

Q5: What is the typical range of values for the midsphere radius?
A: The midsphere radius depends on the size of the polyhedron. For a unit truncated cuboctahedron (edge length = 1), the midsphere radius is approximately 2.5 units.