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Surface to Volume Ratio of Dodecahedron Formula:
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The Surface to Volume Ratio of a Dodecahedron is a geometric property that represents the relationship between the total surface area and the volume of this twelve-faced polyhedron. It's an important parameter in various scientific and engineering applications.
The calculator uses the formula:
Where:
Explanation: This formula calculates the surface area to volume ratio based on the midsphere radius of a regular dodecahedron, incorporating the mathematical constant φ (phi) through the square root of 5.
Details: The surface to volume ratio is crucial in materials science, chemistry, and physics as it affects properties like heat transfer, reaction rates, and mechanical strength. For dodecahedral structures, this ratio is particularly important in crystallography and nanotechnology applications.
Tips: Enter the midsphere radius of the dodecahedron in meters. The value must be positive and non-zero. The calculator will compute the corresponding surface to volume ratio.
Q1: What is a dodecahedron?
A: A dodecahedron is a three-dimensional shape with twelve flat faces, each being a regular pentagon. It is one of the five Platonic solids.
Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that touches the midpoints of all edges of the dodecahedron.
Q3: Why is surface to volume ratio important?
A: This ratio is critical in many physical processes where surface interactions dominate, such as in catalysis, heat exchange, and biological systems.
Q4: What are typical values for this ratio?
A: The ratio varies inversely with size - smaller dodecahedra have higher surface to volume ratios. The exact value depends on the specific dimensions of the dodecahedron.
Q5: Can this formula be used for irregular dodecahedra?
A: No, this specific formula applies only to regular dodecahedra where all faces are identical regular pentagons and all vertices are equivalent.