Volume of Truncated Dodecahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ V = \frac{5}{12} \times (99 + 47\sqrt{5}) \times \left( \frac{12 \times (\sqrt{3} + 6\sqrt{5 + 2\sqrt{5}})}{RA/V \times (99 + 47\sqrt{5})} \right)^3 \]

Volume of Truncated Dodecahedron:

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1. What is the Volume of Truncated Dodecahedron?

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{12} \times (99 + 47\sqrt{5}) \times \left( \frac{12 \times (\sqrt{3} + 6\sqrt{5 + 2\sqrt{5}})}{RA/V \times (99 + 47\sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Truncated Dodecahedron (m³)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: The formula calculates the volume based on the given surface to volume ratio, using mathematical constants and geometric relationships specific to the truncated dodecahedron shape.

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for material estimation, structural analysis, and geometric modeling of truncated dodecahedron shapes in various engineering and architectural applications.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Dodecahedron?
A: A truncated dodecahedron is an Archimedean solid created by truncating the vertices of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

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

Q3: What units should I use?
A: Use consistent units. If surface area is in m² and volume in m³, then surface to volume ratio will be in 1/m.

Q4: Are there limitations to this formula?
A: This formula is specifically designed for perfect truncated dodecahedrons and assumes ideal geometric properties.

Q5: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of values, but extreme values may be limited by floating-point precision.