Pyramidal Height Of Great Stellated Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ h_{Pyramid} = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{6} \cdot \sqrt{\frac{TSA}{15 \cdot \sqrt{5 + 2\sqrt{5}}}} \]

Pyramidal Height (h_Pyramid):

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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 distance from the center of the polyhedron to the apex of one of its pyramidal components.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ h_{Pyramid} = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{6} \cdot \sqrt{\frac{TSA}{15 \cdot \sqrt{5 + 2\sqrt{5}}}} \]

Where:

\( h_{Pyramid} \) — Pyramidal Height (meters)
\( TSA \) — Total Surface Area of the Great Stellated Dodecahedron (m²)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the pyramidal height based on the total surface area of the polyhedron, incorporating mathematical constants and geometric relationships specific to the Great Stellated Dodecahedron.

3. Importance of Pyramidal Height Calculation

Details: Calculating the pyramidal height is essential for understanding the three-dimensional geometry of the Great Stellated Dodecahedron, its spatial proportions, and for various applications in mathematical modeling and geometric analysis.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero for accurate calculation.

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 stellating a regular dodecahedron with triangular pyramids on each face.

Q2: Why is the formula so complex?
A: The formula incorporates mathematical constants and geometric relationships specific to this complex polyhedron, including the golden ratio and square roots.

Q3: What units should I use for the calculation?
A: Use consistent units (typically meters for length and square meters for area). The calculator assumes metric units.

Q4: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this calculation applicable to other polyhedra?
A: No, this specific formula is only valid for the Great Stellated Dodecahedron due to its unique geometric properties.