Volume Of Small Stellated Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ V = \left( \frac{5}{4} \times (7 + 3 \times \sqrt{5}) \right) \times \left( \frac{TSA}{15 \times \sqrt{5 + 2 \times \sqrt{5}}} \right)^{\frac{3}{2}} \]

Volume Of Small Stellated Dodecahedron:

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1. What Is The Volume Of Small Stellated Dodecahedron?

The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron. Its volume represents the total three-dimensional space enclosed by its surface, which is crucial for understanding its geometric properties and applications in various fields.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ V = \left( \frac{5}{4} \times (7 + 3 \times \sqrt{5}) \right) \times \left( \frac{TSA}{15 \times \sqrt{5 + 2 \times \sqrt{5}}} \right)^{\frac{3}{2}} \]

Where:

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

Explanation: This formula derives the volume from the total surface area by establishing a relationship between surface area and the polyhedron's geometric properties.

3. Importance Of Volume Calculation

Details: Calculating the volume of geometric shapes like the Small Stellated Dodecahedron is essential in mathematics, architecture, and engineering for understanding spatial properties, material requirements, and structural characteristics.

4. Using The Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It's a regular star polyhedron with 12 pentagram faces, representing one of the four Kepler-Poinsot solids.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the polyhedron and the mathematical relationship between surface area and volume.

Q3: What are practical applications of this calculation?
A: Applications include mathematical research, architectural design, and understanding geometric properties in various scientific fields.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact when using the precise formula, though computational precision may vary.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron due to its unique geometric properties.