Total Surface Area of Great Stellated Dodecahedron given Volume Calculator

Formula Used:

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

Total Surface Area of Great Stellated Dodecahedron:

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

The Total Surface Area of a Great Stellated Dodecahedron is the total area of all its triangular faces. It is a complex polyhedron with star-shaped faces and is one of the Kepler-Poinsot solids.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the surface area based on the volume of the polyhedron, using mathematical relationships specific to the Great Stellated Dodecahedron's geometry.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is important for understanding the geometric properties of the polyhedron, material requirements for physical models, and for various mathematical and engineering applications involving this specific shape.

4. Using the Calculator

Tips: Enter the volume of the Great 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 Great Stellated Dodecahedron?
A: It is a regular star polyhedron with 12 pentagrammic faces, 30 edges, and 20 vertices. It is one of the four Kepler-Poinsot solids.

Q2: Why is the formula so complex?
A: The complexity arises from the mathematical relationships between volume and surface area in this specific polyhedral shape, which involves irrational numbers and fractional exponents.

Q3: What units should I use?
A: Use consistent units (typically meters for length measurements). The calculator returns surface area in square meters when volume is provided in cubic meters.

Q4: Can this calculator handle very large or very small values?
A: Yes, the calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this formula specific to the Great Stellated Dodecahedron?
A: Yes, this formula is specifically derived for calculating the surface area of a Great Stellated Dodecahedron when its volume is known.