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Circumsphere Radius Formula:
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The circumsphere radius of a truncated icosahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. The truncated icosahedron is an Archimedean solid with 32 faces (12 regular pentagons and 20 regular hexagons), best known as the shape of a soccer ball.
The calculator uses the formula:
Where:
Explanation: This formula derives from the geometric properties of the truncated icosahedron and provides the exact circumsphere radius based on the edge length.
Details: Calculating the circumsphere radius is essential in geometry, materials science, and architecture for understanding the spatial dimensions of this complex polyhedron and its applications in various fields.
Tips: Enter the edge length of the truncated icosahedron in any consistent units. The result will be in the same units. The value must be greater than zero.
Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons), 90 edges, and 60 vertices. It's known for its soccer ball shape.
Q2: How is this formula derived?
A: The formula is derived from the geometric relationships between the edge length and the circumsphere radius in a truncated icosahedron, using trigonometric principles and the golden ratio.
Q3: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for truncated icosahedra. Other polyhedra have different formulas for their circumsphere radii.
Q4: What are practical applications of this calculation?
A: Applications include architectural design, molecular modeling (fullerenes), sports equipment design, and mathematical education.
Q5: How accurate is this calculation?
A: The calculation is mathematically exact when using the formula. The precision depends on the accuracy of the input value and the computational precision.