Pyramidal Height Of Great Stellated Dodecahedron Given Ridge Length Calculator

Formula Used:

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

Pyramidal Height of Great Stellated Dodecahedron:

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1. What is Pyramidal Height of Great Stellated Dodecahedron?

The Pyramidal Height of Great Stellated Dodecahedron is the height of any of the inwards directed tetrahedral pyramids of the Great Stellated Dodecahedron. It is a key geometric measurement in understanding the three-dimensional structure of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( h_{Pyramid} \) — Pyramidal Height of Great Stellated Dodecahedron (m)
\( l_{Ridge} \) — Ridge Length of Great Stellated Dodecahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: The formula calculates the pyramidal height based on the ridge length, incorporating mathematical constants related to the geometry of the great stellated dodecahedron.

3. Importance of Pyramidal Height Calculation

Details: Calculating the pyramidal height is essential for understanding the complete geometric properties of the great stellated dodecahedron, including its volume, surface area, and spatial relationships between different components of the polyhedron.

4. Using the Calculator

Tips: Enter the ridge length in meters. The value must be positive and valid. The calculator will compute the corresponding pyramidal height using the mathematical formula.

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: What units should be used for input?
A: The calculator uses meters as the unit of measurement for both input and output values.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the geometric properties of the Great Stellated Dodecahedron only.

Q4: What is the significance of the mathematical constants in the formula?
A: The constants √3 and √5, along with their combinations, represent fundamental geometric relationships inherent in the structure of the Great Stellated Dodecahedron.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact when using precise values for the constants. The calculator provides results rounded to 6 decimal places for practical use.