Circumsphere Radius of Truncated Icosidodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{31 + (12 \times \sqrt{5})} \times 3 \times (1 + \sqrt{3} + \sqrt{5 + (2 \times \sqrt{5})})}{SA:V \times (19 + (10 \times \sqrt{5}))} \]

Circumsphere Radius of Truncated Icosidodecahedron:

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1. What is the Circumsphere Radius of Truncated Icosidodecahedron?

The Circumsphere Radius of a Truncated Icosidodecahedron is the radius of the sphere that contains the polyhedron in such a way that all vertices are lying on the sphere. It's an important geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{31 + (12 \times \sqrt{5})} \times 3 \times (1 + \sqrt{3} + \sqrt{5 + (2 \times \sqrt{5})})}{SA:V \times (19 + (10 \times \sqrt{5}))} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( SA:V \) — Surface Area to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula relates the circumsphere radius to the surface area to volume ratio through the geometric properties of the truncated icosidodecahedron.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, material science, and architecture for understanding the spatial properties and packing efficiency of this complex polyhedron.

4. Using the Calculator

Tips: Enter the surface area 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 Icosidodecahedron?
A: A truncated icosidodecahedron is an Archimedean solid with 120 vertices, 180 edges, and 62 faces (30 squares, 20 hexagons, and 12 decagons).

Q2: What are typical SA:V values for this shape?
A: The surface area to volume ratio depends on the size of the polyhedron, with smaller sizes having higher SA:V ratios.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect truncated icosidodecahedron, using the precise geometric relationships.

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

Q5: What are practical applications of this calculation?
A: Applications include materials science (nanoparticle characterization), architecture (geometric structures), and mathematical research.