Circumsphere Radius Of Great Stellated Dodecahedron Given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{3} \cdot (3+\sqrt{5})}{4} \cdot \left( \frac{4V}{5(3+\sqrt{5})} \right)^{1/3} \]

Circumsphere Radius of Great Stellated Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 in such a way that all vertices lie on the sphere's surface. It represents the smallest sphere that can completely enclose the Great Stellated Dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{3} \cdot (3+\sqrt{5})}{4} \cdot \left( \frac{4V}{5(3+\sqrt{5})} \right)^{1/3} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of Great Stellated Dodecahedron (m³)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula derives the circumsphere radius from the volume of the Great Stellated Dodecahedron using geometric relationships and mathematical constants.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important for understanding the spatial dimensions of the polyhedron, for geometric modeling, and for applications in crystallography, architecture, and mathematical research involving polyhedral structures.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra, formed by extending the faces of a regular dodecahedron until they intersect.

Q2: How is this different from a regular dodecahedron?
A: The Great Stellated Dodecahedron has star-shaped faces and a more complex structure compared to the regular dodecahedron's pentagonal faces.

Q3: What are typical values for the circumsphere radius?
A: The circumsphere radius depends on the volume. For a given volume, the Great Stellated Dodecahedron will have a larger circumsphere radius than a regular dodecahedron due to its extended structure.

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

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most geometric and mathematical applications.