Total Surface Area Of Small Stellated Dodecahedron Given Volume Calculator

Formula Used:

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

Total Surface Area of Small Stellated Dodecahedron:

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1. What is the Total Surface Area of Small Stellated Dodecahedron?

The Total Surface Area of a Small Stellated Dodecahedron is the total quantity of plane enclosed by the entire surface of this complex polyhedron. It represents the sum of the areas of all its triangular faces.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

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

Where:

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

Explanation: This formula derives the surface area from the volume using geometric relationships specific to the Small Stellated Dodecahedron's structure.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is crucial for various applications including material estimation, structural analysis, and understanding the geometric properties of this complex polyhedron in mathematical and architectural contexts.

4. Using the Calculator

Tips: Enter the volume of the Small Stellated Dodecahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It's a Kepler-Poinsot polyhedron that consists of 12 pentagram faces with the Schläfli symbol {5/2,5}. It's one of the four regular star polyhedra.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometric relationships between volume and surface area in this non-convex polyhedron with self-intersecting faces.

Q3: What are practical applications of this calculation?
A: This calculation is used in mathematical research, architectural design, computer graphics, and understanding complex geometric structures.

Q4: Are there limitations to this formula?
A: The formula assumes a perfect geometric shape and may not account for manufacturing tolerances or material properties in practical applications.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron. Other polyhedra have different volume-to-surface-area relationships.