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Volume of Dodecahedron given Surface to Volume Ratio Calculator

Volume of Dodecahedron Formula:

\[ V = \frac{1}{4} \times (15 + 7\sqrt{5}) \times \left( \frac{12\sqrt{25 + 10\sqrt{5}}}{RA/V \times (15 + 7\sqrt{5})} \right)^3 \]

1/m

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1. What is the Volume of Dodecahedron given Surface to Volume Ratio?

The volume of a dodecahedron can be calculated from its surface to volume ratio using a specific mathematical formula. A dodecahedron is a three-dimensional shape with 12 regular pentagonal faces, 20 vertices, and 30 edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{1}{4} \times (15 + 7\sqrt{5}) \times \left( \frac{12\sqrt{25 + 10\sqrt{5}}}{RA/V \times (15 + 7\sqrt{5})} \right)^3 \]

Where:

Explanation: This formula derives the volume from the surface to volume ratio by reversing the standard volume to surface area relationship for a dodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of a dodecahedron is important in geometry, architecture, material science, and various engineering applications where this specific polyhedral shape is used.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a dodecahedron?
A: A dodecahedron is a regular polyhedron with 12 pentagonal faces, 20 vertices, and 30 edges. It is one of the five Platonic solids.

Q2: What are typical surface to volume ratio values for dodecahedrons?
A: The surface to volume ratio depends on the size of the dodecahedron. Smaller dodecahedrons have higher surface to volume ratios, while larger ones have lower ratios.

Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula applies only to regular dodecahedrons where all faces are identical regular pentagons.

Q4: What are the practical applications of dodecahedron volume calculations?
A: Dodecahedron volume calculations are used in crystallography, architecture, game design, and various scientific simulations.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for regular dodecahedrons, though practical measurements of surface to volume ratio may introduce some error.

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