Edge Length of Small Stellated Dodecahedron given Circumradius Calculator

Formula Used:

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

Edge Length of Small Stellated Dodecahedron:

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1. What is the Edge Length of Small Stellated Dodecahedron given Circumradius?

The edge length of a Small Stellated Dodecahedron given its circumradius is calculated using a specific mathematical formula that relates these two geometric properties of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( l_e \) — Edge Length of Small Stellated Dodecahedron (m)
\( r_c \) — Circumradius of Small Stellated Dodecahedron (m)

Explanation: This formula establishes the precise mathematical relationship between the circumradius and edge length of a Small Stellated Dodecahedron.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from the circumradius is essential for geometric analysis, 3D modeling, and understanding the spatial properties of the Small Stellated Dodecahedron in various applications.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and valid for accurate calculation of the edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: A Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron that represents one of the four regular star polyhedra.

Q2: Why is this formula important in geometry?
A: This formula provides a fundamental relationship between two key geometric properties, enabling accurate dimensional calculations for this specific polyhedron.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron as it's derived from its unique geometric properties.

Q4: What are the practical applications of this calculation?
A: Applications include architectural design, mathematical modeling, computer graphics, and educational demonstrations of geometric principles.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise when correct input values are provided, as it's based on the exact geometric relationship between circumradius and edge length.