Edge Length of Great Stellated Dodecahedron given Circumradius Calculator

Formula Used:

\[ l_e = \frac{4 \times r_c}{\sqrt{3} \times (3 + \sqrt{5})} \]

Edge Length of Great Stellated Dodecahedron:

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

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \frac{4 \times r_c}{\sqrt{3} \times (3 + \sqrt{5})} \]

Where:

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

Explanation: This formula establishes the precise mathematical relationship between the edge length and circumradius of a Great Stellated Dodecahedron, incorporating the mathematical constants √3 and √5.

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 this complex polyhedral shape.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: A Great Stellated Dodecahedron is a Kepler-Poinsot polyhedron that is one of the four regular star polyhedra, formed by extending the faces of a regular dodecahedron.

Q2: What units should I use for the circumradius?
A: The calculator uses meters as the unit of measurement, but you can use any consistent unit as long as you interpret the result in the same unit.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Stellated Dodecahedron. Other polyhedra have different mathematical relationships between edge length and circumradius.

Q4: What is the significance of √3 and √5 in the formula?
A: These irrational numbers appear naturally in the geometry of regular and star polyhedra, particularly those based on the golden ratio and pentagonal symmetry.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Great Stellated Dodecahedron, limited only by the precision of the input values and computational rounding.