Total Surface Area of Great Dodecahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = 15 \times \sqrt{5 - (2 \times \sqrt{5})} \times \left( \frac{15 \times \sqrt{5 - (2 \times \sqrt{5})}}{\frac{5}{4} \times (\sqrt{5} - 1) \times \frac{SA}{V}} \right)^2 \]

Total Surface Area (TSA):

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

The Total Surface Area of a Great Dodecahedron is the total quantity of plane enclosed on the entire surface of this complex polyhedron. It represents the sum of all its face areas.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 15 \times \sqrt{5 - (2 \times \sqrt{5})} \times \left( \frac{15 \times \sqrt{5 - (2 \times \sqrt{5})}}{\frac{5}{4} \times (\sqrt{5} - 1) \times \frac{SA}{V}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area (m²)
\( \frac{SA}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the total surface area based on the given surface to volume ratio, using the mathematical properties of the Great Dodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for understanding the geometric properties of the Great Dodecahedron, material requirements for physical models, and various engineering applications involving this complex shape.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A Great Dodecahedron is one of the four Kepler-Poinsot polyhedra, consisting of 12 pentagonal faces that intersect each other.

Q2: How is this different from a regular dodecahedron?
A: Unlike a regular dodecahedron where faces don't intersect, the Great Dodecahedron has self-intersecting pentagonal faces.

Q3: What are typical surface to volume ratio values?
A: The surface to volume ratio depends on the size of the polyhedron, with smaller objects having higher ratios.

Q4: What units should I use?
A: Use consistent units - surface area in square meters and surface to volume ratio in per meter (1/m).

Q5: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may affect precision.