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The Circumsphere Radius of Rotunda is the radius of the sphere that contains the Rotunda in such a way that all the vertices of the Rotunda are touching the sphere. It represents the smallest sphere that can completely enclose the Rotunda polyhedron.
The calculator uses the formula:
Where:
Explanation: This formula calculates the circumsphere radius based on the height of the Rotunda, incorporating the golden ratio (φ) which is fundamental to the geometry of the Rotunda.
Details: Calculating the circumsphere radius is important in geometry and 3D modeling for determining the bounding sphere of a Rotunda, which is useful in collision detection, spatial analysis, and understanding the spatial properties of this Johnson solid.
Tips: Enter the height of the Rotunda in meters. The value must be positive and greater than zero. The calculator will compute the circumsphere radius using the mathematical relationship between height and circumsphere radius.
Q1: What is a Rotunda in geometry?
A: A Rotunda is a Johnson solid (J6) consisting of 10 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.
Q2: Why does the formula include √5?
A: The square root of 5 appears because the Rotunda's geometry is related to the golden ratio (φ = (1+√5)/2), which is fundamental to its construction.
Q3: Can this calculator be used for other polyhedra?
A: No, this specific formula is only valid for the Rotunda (Johnson solid J6). Other polyhedra have different formulas for their circumsphere radii.
Q4: What are typical values for Rotunda dimensions?
A: The dimensions depend on the specific Rotunda, but the circumsphere radius is always larger than the height due to the polyhedron's geometry.
Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect Rotunda. The accuracy depends on the precision of the input height value.