Volume Of Great Stellated Dodecahedron Given Total Surface Area Calculator

Formula Used:

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

Volume (V):

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1. What Is The Volume Of Great Stellated Dodecahedron?

The Volume of Great Stellated Dodecahedron is the total quantity of three-dimensional space enclosed by the surface of the Great Stellated Dodecahedron. It is a complex polyhedron with star-shaped faces.

2. How Does The Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Great Stellated Dodecahedron (m³)
\( TSA \) — Total Surface Area of Great Stellated Dodecahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives the volume from the total surface area by incorporating geometric constants and mathematical operations specific to the Great Stellated Dodecahedron's structure.

3. Importance Of Volume Calculation

Details: Calculating the volume of complex polyhedra like the Great Stellated Dodecahedron is important in geometry, architectural design, and mathematical modeling, helping understand spatial properties and structural integrity.

4. Using The Calculator

Tips: Enter the total surface area in square meters. The value must be positive. The calculator will compute the volume based on the given formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It is a Kepler-Poinsot polyhedron with star-shaped faces, formed by extending the faces of a regular dodecahedron.

Q2: Why is the formula so complex?
A: The complexity arises from the polyhedron's non-standard shape and the mathematical relationships between its surface area and volume.

Q3: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Great Stellated Dodecahedron due to its unique geometric properties.

Q4: What units should I use?
A: Use consistent units (e.g., meters for length, square meters for area, cubic meters for volume) to ensure accurate results.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact based on the formula, though practical accuracy depends on the precision of the input value.