Circumsphere Radius of Truncated Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{58 + 18\sqrt{5}} \cdot 3(10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{R_{A/V} \cdot (125 + 43\sqrt{5})} \]

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 such that all vertices lie on the sphere's surface. The truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons), best known as the shape of a soccer ball.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{58 + 18\sqrt{5}} \cdot 3(10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{R_{A/V} \cdot (125 + 43\sqrt{5})} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( R_{A/V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{5} \) — Square root of 5 (≈2.23607)
\( \sqrt{3} \) — Square root of 3 (≈1.73205)

Explanation: This formula derives from the geometric properties of the truncated icosahedron and relates the circumsphere radius to the surface-to-volume ratio.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, materials science, and nanotechnology applications where truncated icosahedral structures occur, such as in fullerenes (buckyballs) and certain viral capsids.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

Q2: Where are truncated icosahedra found in nature?
A: This shape appears in carbon fullerenes (C60 molecules), certain viral structures, and architectural designs.

Q3: What is the significance of the circumsphere?
A: The circumsphere represents the smallest sphere that can contain the entire polyhedron, with all vertices touching the sphere's surface.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of a perfect truncated icosahedron.

Q5: Can this calculator handle different units?
A: The calculator uses meters for length units. Convert other units to meters before calculation for accurate results.