Pyramidal Height Of Great Stellated Dodecahedron Calculator

Pyramidal Height Of Great Stellated Dodecahedron Formula:

\[ h_{Pyramid} = \frac{\sqrt{3} \times (3 + \sqrt{5})}{6} \times l_e \]

Pyramidal Height Of Great Stellated Dodecahedron:

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1. What Is The 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 represents the perpendicular distance from the base to the apex of the pyramid formed by the stellation.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ h_{Pyramid} = \frac{\sqrt{3} \times (3 + \sqrt{5})}{6} \times l_e \]

Where:

\( h_{Pyramid} \) — Pyramidal Height of Great Stellated Dodecahedron (m)
\( l_e \) — Edge 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 edge length of the Great Stellated Dodecahedron, incorporating mathematical constants related to its geometric properties.

3. Importance Of Pyramidal Height Calculation

Details: Calculating the pyramidal height is essential for understanding the three-dimensional structure of the Great Stellated Dodecahedron, determining its volume, and analyzing its geometric properties in mathematical and architectural applications.

4. Using The Calculator

Tips: Enter the edge length of the Great Stellated Dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding pyramidal height.

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 this different from a regular dodecahedron?
A: While both are based on dodecahedral symmetry, the Great Stellated Dodecahedron has star-shaped faces and more complex geometry with self-intersecting features.

Q3: What are practical applications of this calculation?
A: This calculation is used in mathematical geometry, architectural design, crystalography, and in creating complex polyhedral models.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Stellated Dodecahedron. Other polyhedra have different formulas for their pyramidal heights.

Q5: What precision should I use for the edge length?
A: For most applications, 4-6 decimal places provide sufficient precision, though the calculator accepts up to 4 decimal places by default.