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The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra. It is formed by stellating the dodecahedron and has 12 pentagrammic faces. The volume calculation helps in understanding its spatial properties and geometric characteristics.
The calculator uses the formula:
Where:
Explanation: This formula calculates the volume based on the pentagram chord length, incorporating the mathematical constant √5 which is fundamental to pentagonal geometry.
Details: Calculating the volume of geometric solids is essential in mathematics, architecture, and engineering for understanding spatial relationships, material requirements, and structural properties.
Tips: Enter the pentagram chord length in meters. The value must be positive and greater than zero for accurate calculation.
Q1: What is a Great Stellated Dodecahedron?
A: It's a regular star polyhedron with 12 pentagram faces, 30 edges, and 20 vertices, formed by extending the faces of a regular dodecahedron.
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 Great Stellated Dodecahedron.
Q3: Why does the formula contain √5?
A: The square root of 5 appears naturally in formulas related to pentagons and pentagrams due to the golden ratio φ = (1+√5)/2, which is fundamental to pentagonal geometry.
Q4: What are the practical applications of this calculation?
A: While primarily mathematical, such calculations find applications in crystallography, architectural design, and the study of geometric patterns in nature.
Q5: How accurate is this calculation?
A: The calculation is mathematically exact when using the formula with precise input values, though practical measurements may introduce some error.