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The Circumsphere Radius of Small Stellated Dodecahedron is the radius of sphere that contains the Small Stellated Dodecahedron in such a way that all vertices are lying on sphere. It is an important geometric property used in three-dimensional geometry and polyhedron analysis.
The calculator uses the formula:
Where:
Explanation: This formula calculates the circumsphere radius based on the pentagram chord length, incorporating the mathematical constant √5 which is fundamental to pentagonal geometry.
Details: Calculating the circumsphere radius is essential for understanding the spatial properties of the Small Stellated Dodecahedron, including its volume, surface area relationships, and its positioning within a circumscribed sphere.
Tips: Enter the Pentagram Chord length in meters. The value must be positive and non-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 that consists of 12 pentagram faces with five pentagrams meeting at each vertex.
Q2: What is the Pentagram Chord in this context?
A: The Pentagram Chord is the distance between any pair of non-adjacent peak vertices of the pentagram corresponding to the Small Stellated Dodecahedron.
Q3: Why is √5 significant in this calculation?
A: √5 appears frequently in formulas related to pentagonal and dodecahedral geometry because it is intrinsically linked to the golden ratio φ = (1+√5)/2.
Q4: What are typical values for the circumsphere radius?
A: The circumsphere radius depends on the scale of the polyhedron. For a unit pentagram chord, the circumsphere radius is approximately 0.5878 meters.
Q5: 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.