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Volume Of Great Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ V = \frac{5}{4} \times (\sqrt{5} - 1) \times \left( \frac{TSA}{15 \times \sqrt{5 - (2 \times \sqrt{5})}} \right)^{\frac{3}{2}} \]

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1. What is the Volume of Great Dodecahedron?

The Great Dodecahedron is one of the Kepler-Poinsot polyhedra. Its volume represents the total three-dimensional space enclosed within its surface. This calculator computes the volume based on the total surface area using the specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{4} \times (\sqrt{5} - 1) \times \left( \frac{TSA}{15 \times \sqrt{5 - (2 \times \sqrt{5})}} \right)^{\frac{3}{2}} \]

Where:

Explanation: This formula derives the volume from the total surface area by incorporating the geometric properties and mathematical constants specific to the Great Dodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes like the Great Dodecahedron is fundamental in mathematics, engineering, and architecture for understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive. The calculator will compute the corresponding volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: The Great Dodecahedron is a regular star polyhedron with 12 pentagonal faces. It is one of the four Kepler-Poinsot solids.

Q2: Why is the formula so complex?
A: The formula involves mathematical constants and operations that capture the unique geometric properties of the Great Dodecahedron, ensuring accurate volume calculation.

Q3: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Great Dodecahedron. Other polyhedra have different formulas for volume calculation.

Q4: What units should I use?
A: The calculator expects the total surface area in square meters and returns the volume in cubic meters. Ensure consistent units for accurate results.

Q5: Is the Great Dodecahedron a common shape?
A: While not as common as simple polyhedra, the Great Dodecahedron is studied in advanced geometry and has applications in mathematical modeling and design.

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