Edge Length of Elongated Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ l_e = \frac{\sqrt{3} \times (3 + \sqrt{5})}{3 \times R_{A/V}} \]

Edge Length (l_e):

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1. What is the Edge Length of Elongated Dodecahedron?

The Edge Length of an Elongated Dodecahedron is the distance between any pair of adjacent peak vertices of the polyhedron. It is a fundamental geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \frac{\sqrt{3} \times (3 + \sqrt{5})}{3 \times R_{A/V}} \]

Where:

\( l_e \) — Edge Length of Elongated Dodecahedron (m)
\( R_{A/V} \) — Surface to Volume Ratio of Elongated Dodecahedron (1/m)
\( \sqrt{3} \) — Square root of 3 (≈1.732)
\( \sqrt{5} \) — Square root of 5 (≈2.236)

Explanation: This formula calculates the edge length based on the surface to volume ratio of the elongated dodecahedron, utilizing mathematical constants and geometric relationships.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the geometric properties of the elongated dodecahedron, including its surface area, volume, and other dimensional characteristics in mathematical modeling and design applications.

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 an Elongated Dodecahedron?
A: An elongated dodecahedron is a polyhedron with 12 faces that has been extended along one axis, creating a shape with specific geometric properties.

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is a critical parameter that influences various physical properties, including heat transfer, chemical reactivity, and structural strength.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size and proportions of the elongated dodecahedron, but typically ranges from very small to relatively large values.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the elongated dodecahedron and may not apply to other polyhedral shapes.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise based on the input values, assuming the elongated dodecahedron follows the defined geometric properties.