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Circumsphere Radius Of Truncated Dodecahedron Given Midsphere Radius Calculator

Formula Used:

\[ r_c = \frac{\sqrt{74 + 30\sqrt{5}} \cdot r_m}{5 + 3\sqrt{5}} \]

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1. What is Circumsphere Radius of Truncated Dodecahedron?

The Circumsphere Radius of a Truncated Dodecahedron is the radius of the sphere that contains the polyhedron in such a way that all its vertices lie on the sphere's surface. It represents the smallest sphere that can completely enclose the truncated dodecahedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_c = \frac{\sqrt{74 + 30\sqrt{5}} \cdot r_m}{5 + 3\sqrt{5}} \]

Where:

Explanation: This formula establishes the precise mathematical relationship between the circumsphere radius and midsphere radius of a truncated dodecahedron, incorporating the golden ratio (√5) which is fundamental to dodecahedral geometry.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometric modeling, architectural design, and materials science where truncated dodecahedrons are used. It helps determine the spatial requirements and bounding dimensions of these complex polyhedral structures.

4. Using the Calculator

Tips: Enter the midsphere radius value in meters. The value must be positive and non-zero. The calculator will compute the corresponding circumsphere radius using the established geometric relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid obtained by truncating the vertices of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

Q2: How is circumsphere radius different from midsphere radius?
A: The circumsphere passes through all vertices of the polyhedron, while the midsphere is tangent to all edges. The circumsphere is always larger than or equal to the midsphere.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, nanotechnology, architectural design, and geometric modeling where precise spatial dimensions of polyhedral structures are required.

Q4: Why does the formula contain √5?
A: The golden ratio (φ = (1+√5)/2) is intrinsically related to regular dodecahedrons and their truncated variants, hence √5 appears in the mathematical relationships between their geometric properties.

Q5: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to truncated dodecahedrons. Other polyhedrons have different mathematical relationships between their circumsphere and midsphere radii.

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