Circumsphere Radius of Great Stellated Dodecahedron Calculator

Formula Used:

\[ r_c = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{4} \cdot l_e \]

Circumsphere Radius of Great Stellated Dodecahedron:

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1. What is the Circumsphere Radius of Great Stellated Dodecahedron?

The Circumsphere Radius of a Great Stellated Dodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. It is a key geometric property that helps in understanding the spatial dimensions and symmetry of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{4} \cdot l_e \]

Where:

\( r_c \) — Circumsphere Radius of Great Stellated Dodecahedron (m)
\( l_e \) — Edge Length of Great Stellated Dodecahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: The formula combines mathematical constants and the edge length to determine the radius of the circumscribed sphere that perfectly encloses the polyhedron.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for geometric analysis, 3D modeling, and understanding the spatial relationships in polyhedral structures. It helps in applications ranging from architectural design to mathematical research.

4. Using the Calculator

Tips: Enter the edge length of the Great Stellated Dodecahedron in meters. The value must be positive and non-zero. The calculator will compute the circumsphere radius using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It is a Kepler-Poinsot polyhedron with star-shaped faces, formed by extending the faces of a regular dodecahedron until they intersect.

Q2: Why is the circumsphere radius important?
A: It defines the smallest sphere that can contain the polyhedron, which is crucial for packaging, containment, and symmetry studies.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Stellated Dodecahedron due to its unique geometric properties.

Q4: What units should I use for the edge length?
A: The calculator uses meters, but you can use any consistent unit as the result will be in the same unit.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the provided formula, with results rounded to 10 decimal places for precision.