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The circumsphere radius of a Small Stellated Dodecahedron is the radius of the sphere that completely contains the polyhedron, with all vertices lying on the surface of the sphere. It is a fundamental geometric property of this complex polyhedral shape.
The calculator uses the mathematical formula:
Where:
Explanation: This formula derives from the geometric properties of the Small Stellated Dodecahedron and relates the circumsphere radius to the total surface area through mathematical constants and square root operations.
Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the polyhedron, for architectural applications, mathematical modeling, and in various fields of geometry and crystallography.
Tips: Enter the total surface area in square meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumsphere radius.
Q1: What is a Small Stellated Dodecahedron?
A: It's a Kepler-Poinsot polyhedron formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron with 12 pentagram faces.
Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Small Stellated Dodecahedron, which involves golden ratio proportions and multiple square root operations.
Q3: What are typical values for the circumsphere radius?
A: The radius depends on the size of the polyhedron. For standard units, it typically ranges from a few centimeters to several meters, depending on the surface area.
Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron. Other polyhedra have different geometric relationships.
Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most practical applications in geometry and design.