Circumsphere Radius of Truncated Icosahedron given Midsphere Radius Calculator

Formula Used:

\[ r_c = \frac{\sqrt{58 + 18\sqrt{5}} \cdot r_m}{3(1 + \sqrt{5})} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Truncated Icosahedron?

The Circumsphere Radius of a Truncated Icosahedron is the radius of the sphere that contains the polyhedron in such a way that all the vertices lie on the sphere's surface. The truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons).

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{58 + 18\sqrt{5}} \cdot r_m}{3(1 + \sqrt{5})} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( r_m \) — Midsphere Radius (m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the circumsphere radius and midsphere radius of a truncated icosahedron, incorporating the golden ratio and square roots.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometry, materials science, and molecular modeling, particularly in understanding the spatial dimensions and packing efficiency of truncated icosahedral structures like fullerenes (buckyballs).

4. Using the Calculator

Tips: Enter the midsphere radius value in meters. The value must be positive and non-zero. The calculator will compute the corresponding circumsphere radius with high precision.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid obtained by truncating the vertices of a regular icosahedron, resulting in 32 faces (12 pentagons and 20 hexagons).

Q2: What are practical applications of this calculation?
A: This calculation is used in molecular modeling (particularly for fullerenes like C60), architectural design, and geometric analysis of polyhedral structures.

Q3: How accurate is this formula?
A: The formula is mathematically exact for perfect truncated icosahedrons and provides precise results when implemented with sufficient numerical precision.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, then convert the result back to your desired unit.

Q5: What is the significance of the mathematical constants in the formula?
A: The constants \( \sqrt{5} \) and the golden ratio \( \phi = \frac{1+\sqrt{5}}{2} \) appear naturally in the geometry of icosahedral structures due to their five-fold symmetry.