Circumsphere Radius of Small Stellated Dodecahedron given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{50+22\sqrt{5}}}{4} \times \left( \frac{4V}{5(7+3\sqrt{5})} \right)^{\frac{1}{3}} \]

Circumsphere Radius of Small Stellated Dodecahedron:

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

The circumsphere radius of a Small 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 of this Kepler-Poinsot solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{50+22\sqrt{5}}}{4} \times \left( \frac{4V}{5(7+3\sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of Small Stellated Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives from the geometric properties of the Small Stellated Dodecahedron, relating its circumradius to its volume 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 polyhedron, its relationship to enclosing spheres, and for applications in geometry, crystallography, and architectural design.

4. Using the Calculator

Tips: Enter the volume of the Small Stellated Dodecahedron in cubic meters. The value must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It is one of the four Kepler-Poinsot solids, formed by extending the faces of a regular dodecahedron until they intersect.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the polyhedron and the mathematical relationships between its volume and circumradius.

Q3: What are typical values for the circumradius?
A: The circumradius depends on the volume. For a unit volume, the circumradius is approximately 0.5-1.0 meters, but this varies with the specific dimensions.

Q4: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Small Stellated Dodecahedron. Other polyhedra have different formulas for their circumradii.

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