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Circumsphere Radius of Truncated Icosidodecahedron given Midsphere Radius Calculator

Formula Used:

\[ r_c = \frac{\sqrt{31 + (12 \times \sqrt{5})} \times r_m}{\sqrt{30 + (12 \times \sqrt{5})}} \]

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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 such that all vertices lie on the sphere's surface. It is a key geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{31 + (12 \times \sqrt{5})} \times r_m}{\sqrt{30 + (12 \times \sqrt{5})}} \]

Where:

Explanation: This formula establishes the mathematical relationship between the circumsphere radius and midsphere radius of a truncated icosidodecahedron, derived from its geometric properties.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions and geometric properties of truncated icosidodecahedrons, which have applications in crystallography, architecture, and mathematical modeling.

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 using the established mathematical relationship.

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 regular hexagons, and 12 regular decagons).

Q2: What is the difference between circumsphere and midsphere?
A: The circumsphere passes through all vertices of the polyhedron, while the midsphere is tangent to all edges of the polyhedron.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated icosidodecahedrons as it's derived from their unique geometric properties.

Q4: What are practical applications of this calculation?
A: This calculation is used in mathematical research, 3D modeling, architectural design, and in understanding the geometric properties of complex polyhedra.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, though practical implementations may have minor rounding errors depending on the precision of input values and computational methods.

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