Ridge Length of Great Stellated Dodecahedron Given Circumradius Calculator

Formula Used:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{4 \times r_c}{\sqrt{3} \times (3+\sqrt{5})} \]

Ridge Length of Great Stellated Dodecahedron:

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1. What is Ridge Length of Great Stellated Dodecahedron?

The Ridge Length of Great Stellated Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Stellated Dodecahedron. It is an important geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{4 \times r_c}{\sqrt{3} \times (3+\sqrt{5})} \]

Where:

\( l_{Ridge} \) — Ridge Length of Great Stellated Dodecahedron (m)
\( r_c \) — Circumradius of Great Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula relates the ridge length to the circumradius through the golden ratio and other geometric constants specific to the Great Stellated Dodecahedron.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for understanding the geometric properties of the Great Stellated Dodecahedron, including its symmetry, proportions, and spatial relationships between vertices.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and non-zero. The calculator will compute the corresponding ridge length using the mathematical relationship between these two geometric properties.

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 circumradius defined for this polyhedron?
A: The circumradius is the radius of the sphere that contains the Great Stellated Dodecahedron such that all vertices lie on the sphere's surface.

Q3: What units should I use for input?
A: The calculator uses meters as the default unit, but you can use any consistent unit system as the relationship is proportional.

Q4: Are there other ways to calculate ridge length?
A: Yes, ridge length can also be calculated from other geometric properties such as edge length or pentagram chord length using different formulas.

Q5: What is the golden ratio's role in this formula?
A: The term (1+√5)/2 represents the golden ratio (φ ≈ 1.618), which appears frequently in the geometry of dodecahedra and related polyhedra.