Circumsphere Radius of Truncated Icosidodecahedron Given Volume Calculator

Formula Used:

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

Circumsphere Radius:

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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 its vertices lie on the sphere's surface. It represents the smallest sphere that can completely enclose the truncated icosidodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of truncated icosidodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives from the geometric properties of the truncated icosidodecahedron, relating its circumsphere radius to its volume through mathematical constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, architecture, and material science for understanding the spatial requirements and bounding characteristics of this complex polyhedral shape.

4. Using the Calculator

Tips: Enter the volume of the truncated icosidodecahedron in cubic meters. 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 62 faces (30 squares, 20 regular hexagons, and 12 regular decagons), 180 edges, and 120 vertices.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the truncated icosidodecahedron, which involves multiple regular polygons and requires mathematical constants like √5 for precise calculation.

Q3: What are practical applications of this calculation?
A: This calculation is used in architectural design, molecular modeling, game development, and any field dealing with complex polyhedral structures.

Q4: How accurate is this formula?
A: The formula is mathematically exact for perfect truncated icosidodecahedrons and provides precise results when correct input values are used.

Q5: Can this calculator handle different units?
A: The calculator expects volume input in cubic meters. For other units, convert to cubic meters first or adjust the result accordingly.