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The Insphere Radius of a Dodecahedron is the radius of the largest sphere that can be contained within the dodecahedron such that the sphere touches all faces of the polyhedron tangentially.
The calculator uses the formula:
Where:
Explanation: This formula establishes the mathematical relationship between the insphere radius and circumsphere radius of a regular dodecahedron using the golden ratio properties inherent in its geometry.
Details: Calculating the insphere radius is essential for understanding the internal geometry of dodecahedrons, which has applications in crystallography, molecular modeling, and architectural design where this polyhedral form is utilized.
Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.
Q1: What is a regular dodecahedron?
A: A regular dodecahedron is a three-dimensional shape with twelve identical regular pentagonal faces, twenty vertices, and thirty edges.
Q2: How does the insphere radius relate to the circumsphere radius?
A: The insphere radius is always smaller than the circumsphere radius, with their ratio being constant for all regular dodecahedrons due to their geometric properties.
Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula applies only to regular dodecahedrons where all faces are identical regular pentagons and all vertices are equivalent.
Q4: What are practical applications of dodecahedron geometry?
A: Dodecahedron geometry is used in various fields including crystallography (certain crystal structures), molecular modeling (fullerenes), and even in some dice designs for games.
Q5: Why does the formula contain √5?
A: The presence of √5 (the square root of 5) relates to the golden ratio (φ = (1+√5)/2), which is intrinsically connected to the geometry of regular pentagons that form the faces of a dodecahedron.