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The Circumsphere Radius of Small Stellated Dodecahedron is the radius of the sphere that contains the Small Stellated Dodecahedron in such a way that all vertices are lying on the sphere. It is an important geometric property of this polyhedron.
The calculator uses the formula:
Where:
Explanation: This formula calculates the circumsphere radius based on the ridge length of the small stellated dodecahedron, incorporating the mathematical constant √5 which is fundamental to pentagonal geometry.
Details: Calculating the circumsphere radius is crucial for understanding the spatial properties and dimensions of the small stellated dodecahedron. It helps in geometric modeling, 3D rendering, and mathematical analysis of this complex polyhedron.
Tips: Enter the ridge length of the small stellated dodecahedron in 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: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron, one of four regular non-convex polyhedra. It consists of 12 pentagram faces with the Schläfli symbol {5/2,5}.
Q2: How is Ridge Length defined?
A: Ridge Length is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Small Stellated Dodecahedron.
Q3: What are typical values for Ridge Length?
A: The ridge length depends on the specific size of the polyhedron. For standard models, it typically ranges from a few centimeters to several meters in practical applications.
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 formulas for calculating their circumsphere radii.
Q5: What is the significance of √5 in this formula?
A: √5 appears frequently in formulas related to pentagonal symmetry and the golden ratio, which are fundamental to the geometry of dodecahedrons and their stellated forms.