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Volume Of Great Stellated Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ V = \frac{5}{4}(3+\sqrt{5}) \times \left( \frac{15\sqrt{5+2\sqrt{5}}}{\frac{5}{4}(3+\sqrt{5}) \times AV} \right)^3 \]

1/m

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1. What is Volume of Great Stellated Dodecahedron?

The Great Stellated Dodecahedron is a Kepler-Poinsot polyhedron with pentagrammic faces. Its volume represents the total three-dimensional space enclosed by its surface, which can be calculated when the surface area to volume ratio is known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{4}(3+\sqrt{5}) \times \left( \frac{15\sqrt{5+2\sqrt{5}}}{\frac{5}{4}(3+\sqrt{5}) \times AV} \right)^3 \]

Where:

Explanation: The formula derives the volume from the known surface area to volume ratio using geometric relationships specific to the Great Stellated Dodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and architecture. For the Great Stellated Dodecahedron, understanding its volume helps in material estimation, structural analysis, and mathematical modeling of complex polyhedra.

4. Using the Calculator

Tips: Enter the surface area to volume ratio in 1/m. The value must be positive and greater than zero. The calculator will compute the corresponding volume of the Great Stellated Dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It's one of the four Kepler-Poinsot solids, formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron with pentagram faces.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Great Stellated Dodecahedron, which involves golden ratio relationships and multiple intersecting planes.

Q3: What are typical values for surface area to volume ratio?
A: The ratio depends on the size of the polyhedron. Smaller polyhedra have higher ratios, while larger ones have lower ratios.

Q4: Can this calculator handle different units?
A: The calculator uses consistent SI units (meters). Ensure your input is in 1/meter units for accurate results.

Q5: What practical applications does this calculation have?
A: This calculation is used in mathematical research, architectural design, and in understanding the properties of complex geometric shapes.

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