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The circumsphere radius of an icosidodecahedron is the radius of the sphere that contains the icosidodecahedron in such a way that all the vertices are lying on the sphere. It's a key geometric property of this Archimedean solid.
The calculator uses the formula:
Where:
Explanation: This formula relates the circumsphere radius directly to the perimeter of the pentagonal faces, incorporating the golden ratio (φ = (1+√5)/2) which is fundamental to the geometry of icosidodecahedrons.
Details: Calculating the circumsphere radius is important in geometry, crystallography, and architectural design where icosidodecahedrons are used. It helps in understanding the spatial dimensions and packing efficiency of these structures.
Tips: Enter the pentagonal face perimeter in meters. The value must be positive and greater than zero.
Q1: What is an icosidodecahedron?
A: An icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 30 identical vertices, and 60 edges.
Q2: Why does the formula include √5?
A: The square root of 5 appears because it's related to the golden ratio, which is fundamental to the geometry of pentagons and appears frequently in the mathematics of icosidodecahedrons.
Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the icosidodecahedron due to its unique combination of triangular and pentagonal faces.
Q4: What are practical applications of this calculation?
A: This calculation is used in molecular modeling, geodesic dome design, and in understanding the structure of certain viral capsids and fullerenes.
Q5: How accurate is this formula?
A: The formula is mathematically exact for a perfect icosidodecahedron, as it's derived from the fundamental geometric properties of this shape.