Circumsphere Radius of Small Stellated Dodecahedron Calculator

Circumsphere Radius Formula:

\[ r_c = \frac{\sqrt{50 + 22\sqrt{5}}}{4} \times l_e \]

Circumsphere Radius (rₑ):

Unit Converter ▲

Unit Converter ▼

From:	To:

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 completely contains the polyhedron, with all vertices lying on the surface of this sphere. It represents the smallest sphere that can enclose the entire shape.

2. How Does the Calculator Work?

The calculator uses the circumsphere radius formula:

\[ r_c = \frac{\sqrt{50 + 22\sqrt{5}}}{4} \times l_e \]

Where:

\( r_c \) — Circumsphere radius
\( l_e \) — Edge length of the Small Stellated Dodecahedron
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the radius of the sphere that circumscribes the Small Stellated Dodecahedron based on its edge length, using the mathematical constant √5.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, 3D modeling, and material science for understanding the spatial requirements and bounding dimensions of this complex polyhedral shape.

4. Using the Calculator

Tips: Enter the edge length of the Small Stellated Dodecahedron in any consistent units. The result will be in the same units. The edge length must be a positive value.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It's a Kepler-Poinsot polyhedron created by extending the faces of a regular dodecahedron until they intersect, forming a star-shaped polyhedron with 12 pentagram faces.

Q2: How accurate is this calculation?
A: The calculation is mathematically exact when using the formula with precise arithmetic operations.

Q3: Can this calculator handle different units?
A: Yes, the calculator works with any consistent unit system. Just ensure you use the same units for input and interpret the output accordingly.

Q4: What's the relationship between edge length and circumsphere radius?
A: They are directly proportional - doubling the edge length will double the circumsphere radius.

Q5: Are there practical applications for this calculation?
A: Yes, in architecture, jewelry design, molecular modeling, and any field working with complex geometric structures.