Circumsphere Radius of Great Dodecahedron given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \times \left( \frac{4V}{5(\sqrt{5} - 1)} \right)^{\frac{1}{3}} \]

Circumsphere Radius of Great Dodecahedron:

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1. What is the Circumsphere Radius of Great Dodecahedron?

The Circumsphere Radius of a Great Dodecahedron is the radius of the sphere that contains the polyhedron in such a way that all vertices lie on the sphere's surface. It is a key geometric property that defines the spatial extent of this complex polyhedral shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_c = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \times \left( \frac{4V}{5(\sqrt{5} - 1)} \right)^{\frac{1}{3}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( V \) — Volume of Great Dodecahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives from the geometric properties of the Great Dodecahedron, relating its volume to the radius of its circumscribed sphere through mathematical constants and operations.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the Great Dodecahedron, which has applications in geometry, architecture, and mathematical modeling of complex structures.

4. Using the Calculator

Tips: Enter the volume of the Great Dodecahedron in cubic meters. The value must be positive and valid. The calculator will compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A Great Dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagonal faces that intersect each other, creating a complex star-shaped polyhedron.

Q2: How is this different from a regular dodecahedron?
A: While both have 12 pentagonal faces, the Great Dodecahedron is a star polyhedron where the faces intersect, whereas a regular dodecahedron is a convex polyhedron.

Q3: What are typical values for the circumsphere radius?
A: The circumsphere radius depends on the volume. For a given volume, the Great Dodecahedron has a larger circumsphere radius compared to many other polyhedra due to its star-shaped nature.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Dodecahedron. Other polyhedra have different mathematical relationships between volume and circumsphere radius.

Q5: What precision does this calculator provide?
A: The calculator provides results with up to 12 decimal places, ensuring high precision for mathematical and engineering applications.