Volume of Elongated Dodecahedron given Surface to Volume Ratio Calculator

Volume of Elongated Dodecahedron Formula:

\[ V = 6 \times \left( \frac{\sqrt{3} \times (3 + \sqrt{5})}{3 \times \frac{A}{V}} \right)^3 \]

Volume of Elongated Dodecahedron:

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1. What is the Volume of Elongated Dodecahedron?

The Volume of Elongated Dodecahedron is the total quantity of three dimensional space enclosed by the surface of the Elongated 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:

\[ V = 6 \times \left( \frac{\sqrt{3} \times (3 + \sqrt{5})}{3 \times \frac{A}{V}} \right)^3 \]

Where:

\( V \) — Volume of Elongated Dodecahedron (m³)
\( \frac{A}{V} \) — Surface to Volume Ratio of Elongated Dodecahedron (1/m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

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

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for various applications including material science, architectural design, packaging optimization, and geometric analysis of complex polyhedra.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is an elongated dodecahedron?
A: An elongated dodecahedron is a polyhedron formed by elongating a regular dodecahedron along one of its symmetry axes, creating a shape with 12 pentagonal faces and additional rectangular faces.

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is important in many physical and chemical processes as it affects properties like heat transfer, reaction rates, and material strength.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the specific dimensions of the elongated dodecahedron. Smaller objects generally have higher surface to volume ratios.

Q4: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for elongated dodecahedra. Other polyhedra have different volume formulas.

Q5: What units should I use for the calculation?
A: The calculator uses meters for length units, resulting in cubic meters for volume. Ensure consistent units throughout your calculations.