Volume of Great Dodecahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ V = \frac{5}{4} \times (\sqrt{5}-1) \times \left( \frac{15 \times \sqrt{5-(2 \times \sqrt{5})}}{\frac{5}{4} \times (\sqrt{5}-1) \times \frac{S}{V}} \right)^3 \]

Volume of Great Dodecahedron:

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1. What is the Volume of Great Dodecahedron?

The Volume of Great Dodecahedron is the total quantity of three dimensional space enclosed by the surface of the Great Dodecahedron. It is a key geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{4} \times (\sqrt{5}-1) \times \left( \frac{15 \times \sqrt{5-(2 \times \sqrt{5})}}{\frac{5}{4} \times (\sqrt{5}-1) \times \frac{S}{V}} \right)^3 \]

Where:

\( V \) — Volume of Great Dodecahedron (m³)
\( \frac{S}{V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume based on the surface to volume ratio, using mathematical constants and geometric relationships specific to the Great Dodecahedron.

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for understanding the spatial properties of the Great Dodecahedron, material requirements, and various engineering applications involving this specific geometric shape.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is important in various physical and engineering contexts, including heat transfer, chemical reactions, and material science.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size and specific dimensions of the Great Dodecahedron, but it generally decreases as the size increases.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect Great Dodecahedron shape and may not account for manufacturing tolerances or material imperfections in real-world applications.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Great Dodecahedron. Other polyhedra have different geometric properties and require different formulas.