Ridge Length Of Small Stellated Dodecahedron Given Circumradius Calculator

Formula Used:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{4 \times r_c}{\sqrt{50 + 22\sqrt{5}}} \]

Ridge Length of Small Stellated Dodecahedron:

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1. What is the Ridge Length of Small Stellated Dodecahedron?

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{4 \times r_c}{\sqrt{50 + 22\sqrt{5}}} \]

Where:

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

Explanation: This formula calculates the ridge length based on the circumradius of the polyhedron, using the golden ratio and specific geometric relationships.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is important in geometric analysis, architectural design, and mathematical modeling of polyhedra. It helps in understanding the spatial relationships and proportions within the Small Stellated Dodecahedron.

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.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron, one of four regular non-convex polyhedra. It is formed by extending the faces of a regular dodecahedron.

Q2: What is the significance of the golden ratio in this formula?
A: The golden ratio (φ = (1+√5)/2) appears frequently in the geometry of regular polyhedra, particularly those with pentagonal symmetry like the dodecahedron and its stellations.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact when using precise values. The calculator provides results rounded to 10 decimal places for practical use.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron. Other polyhedra have different geometric relationships and require different formulas.

Q5: What are typical values for the circumradius?
A: The circumradius depends on the specific size of the polyhedron. For a unit Small Stellated Dodecahedron, the circumradius would be approximately 0.5-1.0 meters, but actual values can vary based on the scale of the object.