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The Midsphere Radius of a Truncated Dodecahedron is the radius of the sphere that is tangent to all the edges of the polyhedron. It lies midway between the inscribed and circumscribed spheres.
The calculator uses the formula:
Where:
Explanation: This formula establishes the mathematical relationship between the midsphere radius and circumsphere radius of a truncated dodecahedron, incorporating the golden ratio through the √5 terms.
Details: Calculating the midsphere radius is important in geometric modeling, crystallography, and architectural design where precise spatial relationships between polyhedral elements are required.
Tips: Enter the circumsphere radius in meters. The value must be positive and non-zero. The calculator will compute the corresponding midsphere radius.
Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid obtained by cutting the corners of a regular dodecahedron, resulting in 20 triangular faces and 12 decagonal faces.
Q2: How is the midsphere different from circumsphere and insphere?
A: The circumsphere passes through all vertices, the insphere is tangent to all faces, while the midsphere is tangent to all edges of the polyhedron.
Q3: What are practical applications of this calculation?
A: This calculation is used in molecular modeling, geodesic dome design, and the study of crystal structures in materials science.
Q4: Why does the formula contain √5 terms?
A: The √5 terms appear because the dodecahedron and its truncated form are closely related to the golden ratio (φ = (1+√5)/2), which is fundamental to their geometry.
Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated dodecahedra. Other polyhedra have different relationships between their midsphere and circumsphere radii.