Edge Length of Rhombic Dodecahedron given Volume Calculator

Formula Used:

\[ l_e = \left( \frac{9 \times V}{16 \times \sqrt{3}} \right)^{1/3} \]

Edge Length of Rhombic Dodecahedron (lₑ):

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1. What is the Edge Length of Rhombic Dodecahedron?

The edge length of a rhombic dodecahedron is the length of any of the edges of this polyhedron or the distance between any pair of adjacent vertices. It is a fundamental geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \left( \frac{9 \times V}{16 \times \sqrt{3}} \right)^{1/3} \]

Where:

\( l_e \) — Edge length of the rhombic dodecahedron (m)
\( V \) — Volume of the rhombic dodecahedron (m³)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)

Explanation: This formula calculates the edge length from the given volume by applying the inverse relationship between volume and edge length in a rhombic dodecahedron.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the geometric properties of the rhombic dodecahedron, including surface area calculations, spatial analysis, and applications in crystallography and material science.

4. Using the Calculator

Tips: Enter the volume of the rhombic dodecahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a rhombic dodecahedron?
A: A rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces, 14 vertices, and 24 edges. It is a Catalan solid and the dual polyhedron of the cuboctahedron.

Q2: Why is this specific formula used?
A: This formula is derived from the geometric relationship between the volume and edge length of a rhombic dodecahedron, ensuring accurate calculations based on its specific properties.

Q3: What units should be used for input?
A: The calculator expects volume input in cubic meters (m³), and the result will be in meters (m). Consistent units must be maintained for accurate results.

Q4: Can this calculator handle very large or very small values?
A: Yes, the calculator can process a wide range of positive values, though extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Are there other ways to calculate edge length?
A: Yes, the edge length can also be calculated from other geometric properties such as surface area or using trigonometric relationships, but this calculator specifically uses the volume-based formula.