Ridge Length of Great Stellated Dodecahedron given Pyramidal Height Calculator

Formula Used:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3+\sqrt{5})} \]

Ridge Length of Great Stellated Dodecahedron:

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1. What is the 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:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3+\sqrt{5})} \]

Where:

\( l_{Ridge} \) — Ridge Length of Great Stellated Dodecahedron (m)
\( h_{Pyramid} \) — Pyramidal Height 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 calculates the ridge length based on the pyramidal height using the golden ratio and geometric relationships specific to the great stellated dodecahedron.

3. Importance of Ridge Length Calculation

Details: Accurate calculation of ridge length is crucial for understanding the geometric properties of the great stellated dodecahedron, its symmetry, and for applications in mathematical modeling and architectural design.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the great stellated dodecahedron?
A: The great stellated dodecahedron is one of the Kepler-Poinsot polyhedra, featuring star-shaped faces and complex geometric properties.

Q2: Why does the formula include √5 and √3?
A: These square roots appear due to the golden ratio (φ = (1+√5)/2) and the triangular geometry involved in the polyhedron's structure.

Q3: What units should I use for input?
A: The calculator uses meters as the unit of measurement, but the formula works with any consistent unit system.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the great stellated dodecahedron.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the great stellated dodecahedron due to its unique geometric properties.