1. What is Midsphere Radius of Truncated Icosahedron?

The midsphere radius of a truncated icosahedron is the radius of the sphere that touches all the edges of the polyhedron. For a truncated icosahedron, this sphere is tangent to each edge at its midpoint.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_m \) — Midsphere radius (m)
• \( RA/V \) — Surface to volume ratio (1/m)
• \( \sqrt{5} \) — Square root of 5 (≈2.23607)
• \( \sqrt{3} \) — Square root of 3 (≈1.73205)

Explanation: This formula calculates the midsphere radius based on the surface to volume ratio of a truncated icosahedron, incorporating the mathematical constants specific to this polyhedron's geometry.

3. Importance of Midsphere Radius Calculation

Details: The midsphere radius is important in geometry and materials science for understanding the spatial properties of truncated icosahedral structures, which appear in various natural and synthetic forms including fullerenes (buckyballs).

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons), 90 edges, and 60 vertices. It's the shape of a soccer ball.

Q2: Why is this specific formula used?
A: The formula incorporates the mathematical constants and relationships specific to the geometry of truncated icosahedrons, ensuring accurate calculation of the midsphere radius.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the truncated icosahedron. Smaller structures have higher ratios, while larger ones have lower ratios.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to truncated icosahedrons. Other polyhedra have different geometric relationships and require different formulas.

Q5: What units should I use?
A: Use consistent units - typically meters for length and 1/m for surface to volume ratio. The calculator will return the midsphere radius in the same length units.