Surface to Volume Ratio of Rhombic Dodecahedron Calculator

Surface to Volume Ratio Formula:

\[ \text{Surface to Volume Ratio} = \frac{9\sqrt{2}}{2\sqrt{3} \times \text{Edge Length}} \]

Surface to Volume Ratio:

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1. What is the Surface to Volume Ratio of Rhombic Dodecahedron?

The Surface to Volume Ratio of a Rhombic Dodecahedron is a geometric property that represents the relationship between the total surface area and the volume of this polyhedron. It's an important parameter in materials science, chemistry, and physics where surface interactions are significant.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{9\sqrt{2}}{2\sqrt{3} \times \text{Edge Length}} \]

Where:

Edge Length — The length of any edge of the rhombic dodecahedron (in meters)
√2 — Square root of 2 (approximately 1.4142)
√3 — Square root of 3 (approximately 1.7321)

Explanation: This formula calculates how much surface area is available per unit volume of the rhombic dodecahedron, which decreases as the size of the polyhedron increases.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in many scientific applications. In materials science, it affects reactivity and dissolution rates. In biology, it influences nutrient absorption and heat exchange. In chemistry, it impacts catalytic activity and reaction rates.

4. Using the Calculator

Tips: Enter the edge length of the rhombic dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio in reciprocal meters (m⁻¹).

5. Frequently Asked Questions (FAQ)

Q1: What is a rhombic dodecahedron?
A: A rhombic dodecahedron is a polyhedron with 12 congruent rhombic faces. It's a Catalan solid and the dual polyhedron of the cuboctahedron.

Q2: Why does the surface to volume ratio decrease with increasing size?
A: As the size increases, volume grows faster than surface area (volume ∝ length³, surface area ∝ length²), resulting in a lower ratio.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystal growth studies, nanoparticle research, packaging optimization, and in understanding biological structures with similar geometry.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the rhombic dodecahedron. Other polyhedra have different surface to volume ratio formulas.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect rhombic dodecahedron. Real-world applications may require adjustments for surface roughness or other factors.