Total Surface Area Of Great Stellated Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

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

Total Surface Area of Great Stellated Dodecahedron:

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1. What is the Great Stellated Dodecahedron?

The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra, formed by extending the faces of a regular dodecahedron until they intersect. It has 12 pentagrammic faces and is known for its star-like appearance.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area of Great Stellated Dodecahedron (m²)
\( AV \) — SA:V of Great Stellated Dodecahedron (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the surface area to volume ratio of the polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is important in various fields including mathematics, architecture, material science, and 3D modeling. For the Great Stellated Dodecahedron, understanding its surface properties helps in studying its geometric characteristics and applications in design.

4. Using the Calculator

Tips: Enter the surface area to volume ratio (SA:V) value in 1/m. The value must be greater than 0 for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It's a regular star polyhedron with 12 pentagram faces, 30 edges, and 12 vertices, representing one of the four Kepler-Poinsot solids.

Q2: How is this different from a regular dodecahedron?
A: While both have 12 faces, the Great Stellated Dodecahedron has star-shaped (pentagram) faces instead of pentagonal faces, creating a more complex structure.

Q3: What are practical applications of this calculation?
A: This calculation is useful in mathematical research, architectural design, crystal structure analysis, and creating complex 3D models.

Q4: Why is the surface area to volume ratio important?
A: The SA:V ratio is crucial in many scientific fields as it affects properties like heat transfer, chemical reaction rates, and structural strength.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived specifically for the Great Stellated Dodecahedron based on its unique geometric properties.