Surface to Volume Ratio of Small Stellated Dodecahedron Calculator

Surface to Volume Ratio Formula:

\[ SA:V = \frac{15 \times \sqrt{5 + 2 \times \sqrt{5}}}{\frac{5}{4} \times (7 + 3 \times \sqrt{5})} \times \frac{1}{l_e} \]

Surface to Volume Ratio (SA:V):

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1. What is Surface to Volume Ratio of Small Stellated Dodecahedron?

The surface to volume ratio (SA:V) of a Small Stellated Dodecahedron is the numerical ratio of its total surface area to its volume. It's an important geometric property that describes how much surface area is available per unit volume of the shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ SA:V = \frac{15 \times \sqrt{5 + 2 \times \sqrt{5}}}{\frac{5}{4} \times (7 + 3 \times \sqrt{5})} \times \frac{1}{l_e} \]

Where:

\( l_e \) — Edge length of the Small Stellated Dodecahedron (meters)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the ratio by dividing the constant surface area coefficient by the edge length, where the constant represents the mathematical relationship specific to the Small Stellated Dodecahedron geometry.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and physics. It helps understand properties like diffusion rates, heat transfer, and chemical reactivity that depend on the relationship between surface area and volume.

4. Using the Calculator

Tips: Enter the edge length of the Small Stellated Dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio in units of per meter (m⁻¹).

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: A Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron that represents one of the four regular star polyhedra, formed by extending the faces of a regular dodecahedron.

Q2: Why is the surface to volume ratio important?
A: This ratio is fundamental in understanding how geometric properties scale and is particularly important in nanoscale materials where surface effects dominate bulk properties.

Q3: What units are used for the calculation?
A: The edge length should be in meters, and the resulting surface to volume ratio will be in per meter (m⁻¹) units.

Q4: Can this calculator handle very small edge lengths?
A: Yes, the calculator can handle edge lengths as small as 0.0001 meters, but extremely small values may approach computational limits.

Q5: How accurate is the calculation?
A: The calculation uses precise mathematical constants and functions, providing results accurate to 12 decimal places for most practical applications.