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The Pyramidal Height of Small Stellated Dodecahedron is the height of any of the inwards directed tetrahedral pyramids of the Small Stellated Dodecahedron. It represents the perpendicular distance from the base of the pyramid to its apex within this complex polyhedral structure.
The calculator uses the mathematical formula:
Where:
Details: The formula derives from the geometric properties of the Small Stellated Dodecahedron, incorporating the golden ratio (φ) and its relationship with the pentagonal faces and pyramidal structures that characterize this Kepler-Poinsot polyhedron.
Tips: Enter the total surface area in square meters. The value must be positive and non-zero. The calculator will compute the corresponding pyramidal height based on the mathematical relationship between surface area and pyramidal dimensions.
Q1: What is a Small Stellated Dodecahedron?
A: It's one of the four Kepler-Poinsot polyhedra, formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron with 12 pentagrammic faces.
Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry involving the golden ratio and the relationships between the various dimensions in this non-convex polyhedron.
Q3: What are typical values for pyramidal height?
A: The pyramidal height depends on the overall size of the polyhedron. For standard-sized models, it typically ranges from a few centimeters to several meters, proportional to the total surface area.
Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron due to its unique geometric properties and symmetry group.
Q5: What practical applications does this calculation have?
A: While primarily theoretical, such calculations are important in mathematical research, architectural design, and the study of polyhedral geometry and symmetry.