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Circumsphere Radius of Great Stellated Dodecahedron given Ridge Length Calculator

Formula Used:

\[ r_c = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{2} \cdot \frac{1 \cdot l_{Ridge}}{1 + \sqrt{5}} \]

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1. What is Circumsphere Radius of Great Stellated Dodecahedron?

The Circumsphere Radius of a Great Stellated Dodecahedron is the radius of the sphere that contains the polyhedron in such a way that all vertices are lying on the sphere. It is an important geometric property in polyhedral studies.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{2} \cdot \frac{1 \cdot l_{Ridge}}{1 + \sqrt{5}} \]

Where:

Explanation: This formula calculates the circumsphere radius based on the ridge length of the Great Stellated Dodecahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial properties of the Great Stellated Dodecahedron, its relationship with other geometric elements, and its applications in various mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the Ridge Length in meters. The value must be positive and valid. The calculator will compute the Circumsphere Radius based on the provided input.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: The Great Stellated Dodecahedron is a Kepler-Poinsot polyhedron, one of four regular star polyhedra. It is composed of 12 intersecting pentagram faces.

Q2: How is Ridge Length defined?
A: Ridge Length is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Stellated Dodecahedron.

Q3: What are typical values for Circumsphere Radius?
A: The Circumsphere Radius depends on the Ridge Length. For standard measurements, it typically ranges proportionally based on the input ridge length.

Q4: Are there limitations to this calculation?
A: This formula is specifically designed for the Great Stellated Dodecahedron and assumes perfect geometric construction. It may not apply to other polyhedra.

Q5: What units should be used?
A: The calculator uses meters for both input and output, but any consistent unit system can be used as long as the same unit is maintained throughout.

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