Circumsphere Radius of Icosidodecahedron Given Midsphere Radius Calculator

Formula Used:

\[ r_c = \frac{1 + \sqrt{5}}{\sqrt{5 + 2\sqrt{5}}} \times r_m \]

Circumsphere Radius of Icosidodecahedron (r꜀):

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1. What is the Circumsphere Radius of Icosidodecahedron?

The Circumsphere Radius of an Icosidodecahedron is the radius of the sphere that contains the polyhedron in such a way that all its vertices lie on the surface of this sphere. It is a fundamental geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_c = \frac{1 + \sqrt{5}}{\sqrt{5 + 2\sqrt{5}}} \times r_m \]

Where:

\( r_c \) — Circumsphere Radius of Icosidodecahedron (m)
\( r_m \) — Midsphere Radius of Icosidodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

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

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the icosidodecahedron, its packing efficiency, and its applications in various fields including 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 automatically compute the corresponding circumsphere radius using the precise mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is an Icosidodecahedron?
A: An icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 30 vertices, and 60 edges.

Q2: How is the circumsphere radius different from the midsphere radius?
A: The circumsphere passes through all vertices, while the midsphere is tangent to all edges of the polyhedron.

Q3: What are typical values for these radii?
A: For a standard icosidodecahedron with edge length 1, the midsphere radius is approximately 1.5 and the circumsphere radius is approximately 1.618.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the icosidodecahedron due to its unique geometric properties.

Q5: What practical applications does this calculation have?
A: This calculation is used in geometric modeling, architectural design, molecular structures, and mathematical research involving polyhedral geometry.