Volume Of Great Stellated Dodecahedron Given Pyramidal Height Calculator

Formula Used:

\[ V = \frac{5}{4} \times (3+\sqrt{5}) \times \left(\frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3+\sqrt{5})}\right)^3 \]

Volume of Great Stellated Dodecahedron:

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1. What is the Volume of Great Stellated Dodecahedron?

The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra, consisting of 12 intersecting pentagram faces. Its volume represents the total three-dimensional space enclosed by its surface.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{4} \times (3+\sqrt{5}) \times \left(\frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3+\sqrt{5})}\right)^3 \]

Where:

\( V \) — Volume of Great Stellated Dodecahedron (m³)
\( h_{Pyramid} \) — 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 volume based on the pyramidal height, which is the height of the tetrahedral pyramids that form the stellation.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, physics, engineering, and architecture. For the Great Stellated Dodecahedron, volume calculation helps in understanding its spatial properties and applications in various fields.

4. Using the Calculator

Tips: Enter the pyramidal height in meters. The value must be positive and greater than zero. The calculator will compute the volume using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It's a regular star polyhedron with 12 pentagram faces that intersect each other, forming a complex three-dimensional shape.

Q2: What is pyramidal height in this context?
A: Pyramidal height refers to the height of the tetrahedral pyramids that extend inward from the faces of the dodecahedron to form the stellation.

Q3: What are typical values for pyramidal height?
A: The pyramidal height depends on the specific dimensions of the dodecahedron and can vary based on the scale of the model.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is designed only for the Great Stellated Dodecahedron. Other polyhedra have different volume formulas.

Q5: What are practical applications of this calculation?
A: This calculation is used in mathematical research, geometric modeling, architectural design, and educational contexts to understand the properties of complex polyhedra.