Edge Length of Rhombic Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ l_e = \frac{9\sqrt{2}}{2\sqrt{3} \times R_{A/V}} \]

Edge Length (l_e):

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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:

\[ l_e = \frac{9\sqrt{2}}{2\sqrt{3} \times R_{A/V}} \]

Where:

\( l_e \) — Edge length of rhombic dodecahedron (m)
\( R_{A/V} \) — Surface to volume ratio (1/m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{3} \) — Square root of 3 (approximately 1.7321)

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

3. Importance of Edge Length Calculation

Details: Calculating the edge length is crucial for understanding the geometric properties of rhombic dodecahedrons, which are used in crystallography, material science, and architectural design. It helps in determining volume, surface area, and other dimensional characteristics.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and non-zero. The calculator will compute the corresponding edge length in meters.

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: What are typical applications of rhombic dodecahedrons?
A: They are used in crystallography (as the shape of the Brillouin zone of face-centered cubic crystals), in geodesic domes, and in various engineering and architectural structures.

Q3: How is surface to volume ratio related to edge length?
A: For a given shape, the surface to volume ratio is inversely proportional to the size (edge length) of the object. Smaller objects have higher surface to volume ratios.

Q4: What are the units for edge length and surface to volume ratio?
A: Edge length is typically measured in meters (m), while surface to volume ratio is measured in reciprocal meters (1/m).

Q5: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to rhombic dodecahedrons. Other polyhedrons have different relationships between edge length and surface to volume ratio.