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Pyramidal Height Of Great Dodecahedron Given Circumsphere Radius Calculator

Formula Used:

\[ h_{Pyramid} = \frac{\sqrt{3} \cdot (3 - \sqrt{5})}{6} \cdot \frac{4 \cdot r_c}{\sqrt{10 + (2 \cdot \sqrt{5})}} \]

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1. What is the Pyramidal Height of Great Dodecahedron?

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h_{Pyramid} = \frac{\sqrt{3} \cdot (3 - \sqrt{5})}{6} \cdot \frac{4 \cdot r_c}{\sqrt{10 + (2 \cdot \sqrt{5})}} \]

Where:

Explanation: This formula calculates the pyramidal height based on the circumsphere radius, incorporating mathematical constants and geometric relationships specific to the Great Dodecahedron.

3. Importance of Pyramidal Height Calculation

Details: Calculating the pyramidal height is essential for understanding the internal structure of the Great Dodecahedron, its volume properties, and its geometric relationships with other polyhedral measurements.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and non-zero. The calculator will compute the corresponding pyramidal height using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: The Great Dodecahedron is one of the Kepler-Poinsot polyhedra, consisting of 12 pentagonal faces that intersect each other, creating a complex star-shaped polyhedron.

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the Great Dodecahedron such that all vertices lie on the sphere's surface.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the given formula, with results rounded to 12 decimal places for practical use.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Dodecahedron due to its unique geometric properties.

Q5: What are typical values for pyramidal height?
A: The pyramidal height varies with the size of the polyhedron but maintains a constant proportional relationship with the circumsphere radius.

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