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The circumsphere radius of a Great Stellated Dodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere. It is a key geometric property used in three-dimensional geometry and polyhedron analysis.
The calculator uses the formula:
Where:
Explanation: The formula relates the circumsphere radius to the total surface area through mathematical constants and square root operations specific to the geometry of the Great Stellated Dodecahedron.
Details: Calculating the circumsphere radius is important for understanding the spatial dimensions of the polyhedron, its relationship to enclosing spheres, and for applications in computational geometry and 3D modeling.
Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the circumsphere radius based on the provided surface area.
Q1: What is a Great Stellated Dodecahedron?
A: It is a Kepler-Poinsot polyhedron with star-shaped faces, formed by extending the faces of a regular dodecahedron.
Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Great Stellated Dodecahedron, which involves golden ratio relationships and multiple square roots.
Q3: What are typical values for the circumsphere radius?
A: The radius depends on the size of the polyhedron. For a polyhedron with TSA of 100m², the circumsphere radius would be approximately [calculated value] meters.
Q4: Can this calculator be used for other polyhedra?
A: No, this formula is specific to the Great Stellated Dodecahedron. Other polyhedra have different formulas for their circumsphere radii.
Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the provided formula. Accuracy depends on the precision of the input value and the computational precision of the system.